\(p\)-adic Capped Absolute Elements

Elements of \(p\)-adic Rings with Absolute Precision Cap

AUTHORS:

  • David Roe

  • Genya Zaytman: documentation

  • David Harvey: doctests

class sage.rings.padics.padic_capped_absolute_element.CAElement[source]

Bases: pAdicTemplateElement

add_bigoh(absprec)[source]

Return a new element with absolute precision decreased to absprec. The precision never increases.

INPUT:

  • absprec – integer or infinity

OUTPUT: self with precision set to the minimum of self’s precision and prec

EXAMPLES:

sage: R = Zp(7,4,'capped-abs','series'); a = R(8); a.add_bigoh(1)
1 + O(7)

sage: k = ZpCA(3,5)
sage: a = k(41); a
2 + 3 + 3^2 + 3^3 + O(3^5)
sage: a.add_bigoh(7)
2 + 3 + 3^2 + 3^3 + O(3^5)
sage: a.add_bigoh(3)
2 + 3 + 3^2 + O(3^3)

>>> from sage.all import *
>>> R = Zp(Integer(7),Integer(4),'capped-abs','series'); a = R(Integer(8)); a.add_bigoh(Integer(1))
1 + O(7)

>>> k = ZpCA(Integer(3),Integer(5))
>>> a = k(Integer(41)); a
2 + 3 + 3^2 + 3^3 + O(3^5)
>>> a.add_bigoh(Integer(7))
2 + 3 + 3^2 + 3^3 + O(3^5)
>>> a.add_bigoh(Integer(3))
2 + 3 + 3^2 + O(3^3)
is_equal_to(_right, absprec=None)[source]

Determine whether the inputs are equal modulo \(\pi^{\mbox{absprec}}\).

INPUT:

  • right – a \(p\)-adic element with the same parent

  • absprec – integer, infinity, or None

EXAMPLES:

sage: R = ZpCA(2, 6)
sage: R(13).is_equal_to(R(13))
True
sage: R(13).is_equal_to(R(13+2^10))
True
sage: R(13).is_equal_to(R(17), 2)
True
sage: R(13).is_equal_to(R(17), 5)
False
sage: R(13).is_equal_to(R(13+2^10),absprec=10)
Traceback (most recent call last):
...
PrecisionError: elements not known to enough precision

>>> from sage.all import *
>>> R = ZpCA(Integer(2), Integer(6))
>>> R(Integer(13)).is_equal_to(R(Integer(13)))
True
>>> R(Integer(13)).is_equal_to(R(Integer(13)+Integer(2)**Integer(10)))
True
>>> R(Integer(13)).is_equal_to(R(Integer(17)), Integer(2))
True
>>> R(Integer(13)).is_equal_to(R(Integer(17)), Integer(5))
False
>>> R(Integer(13)).is_equal_to(R(Integer(13)+Integer(2)**Integer(10)),absprec=Integer(10))
Traceback (most recent call last):
...
PrecisionError: elements not known to enough precision
is_zero(absprec=None)[source]

Determine whether this element is zero modulo \(\pi^{\mbox{absprec}}\).

If absprec is None, returns True if this element is indistinguishable from zero.

INPUT:

  • absprec – integer, infinity, or None

EXAMPLES:

sage: R = ZpCA(17, 6)
sage: R(0).is_zero()
True
sage: R(17^6).is_zero()
True
sage: R(17^2).is_zero(absprec=2)
True
sage: R(17^6).is_zero(absprec=10)
Traceback (most recent call last):
...
PrecisionError: not enough precision to determine if element is zero

>>> from sage.all import *
>>> R = ZpCA(Integer(17), Integer(6))
>>> R(Integer(0)).is_zero()
True
>>> R(Integer(17)**Integer(6)).is_zero()
True
>>> R(Integer(17)**Integer(2)).is_zero(absprec=Integer(2))
True
>>> R(Integer(17)**Integer(6)).is_zero(absprec=Integer(10))
Traceback (most recent call last):
...
PrecisionError: not enough precision to determine if element is zero
polynomial(var='x')[source]

Return a polynomial over the base ring that yields this element when evaluated at the generator of the parent.

INPUT:

  • var – string; the variable name for the polynomial

EXAMPLES:

sage: # needs sage.libs.flint
sage: R.<a> = ZqCA(5^3)
sage: a.polynomial()
(1 + O(5^20))*x + O(5^20)
sage: a.polynomial(var='y')
(1 + O(5^20))*y + O(5^20)
sage: (5*a^2 + R(25, 4)).polynomial()
(5 + O(5^4))*x^2 + O(5^4)*x + 5^2 + O(5^4)

>>> from sage.all import *
>>> # needs sage.libs.flint
>>> R = ZqCA(Integer(5)**Integer(3), names=('a',)); (a,) = R._first_ngens(1)
>>> a.polynomial()
(1 + O(5^20))*x + O(5^20)
>>> a.polynomial(var='y')
(1 + O(5^20))*y + O(5^20)
>>> (Integer(5)*a**Integer(2) + R(Integer(25), Integer(4))).polynomial()
(5 + O(5^4))*x^2 + O(5^4)*x + 5^2 + O(5^4)
precision_absolute()[source]

The absolute precision of this element.

This is the power of the maximal ideal modulo which this element is defined.

EXAMPLES:

sage: R = Zp(7,4,'capped-abs'); a = R(7); a.precision_absolute()
4

>>> from sage.all import *
>>> R = Zp(Integer(7),Integer(4),'capped-abs'); a = R(Integer(7)); a.precision_absolute()
4
precision_relative()[source]

The relative precision of this element.

This is the power of the maximal ideal modulo which the unit part of this element is defined.

EXAMPLES:

sage: R = Zp(7,4,'capped-abs'); a = R(7); a.precision_relative()
3

>>> from sage.all import *
>>> R = Zp(Integer(7),Integer(4),'capped-abs'); a = R(Integer(7)); a.precision_relative()
3
unit_part()[source]

Return the unit part of this element.

EXAMPLES:

sage: R = Zp(17,4,'capped-abs', 'val-unit')
sage: a = R(18*17)
sage: a.unit_part()
18 + O(17^3)
sage: type(a)
<class 'sage.rings.padics.padic_capped_absolute_element.pAdicCappedAbsoluteElement'>
sage: R(0).unit_part()
O(17^0)

>>> from sage.all import *
>>> R = Zp(Integer(17),Integer(4),'capped-abs', 'val-unit')
>>> a = R(Integer(18)*Integer(17))
>>> a.unit_part()
18 + O(17^3)
>>> type(a)
<class 'sage.rings.padics.padic_capped_absolute_element.pAdicCappedAbsoluteElement'>
>>> R(Integer(0)).unit_part()
O(17^0)
val_unit()[source]

Return a 2-tuple, the first element set to the valuation of this element, and the second to the unit part of this element.

For a zero element, the unit part is O(p^0).

EXAMPLES:

sage: R = ZpCA(5)
sage: a = R(75, 6); b = a - a
sage: a.val_unit()
(2, 3 + O(5^4))
sage: b.val_unit()
(6, O(5^0))

>>> from sage.all import *
>>> R = ZpCA(Integer(5))
>>> a = R(Integer(75), Integer(6)); b = a - a
>>> a.val_unit()
(2, 3 + O(5^4))
>>> b.val_unit()
(6, O(5^0))
class sage.rings.padics.padic_capped_absolute_element.ExpansionIter[source]

Bases: object

An iterator over a \(p\)-adic expansion.

This class should not be instantiated directly, but instead using expansion().

INPUT:

  • elt – the \(p\)-adic element

  • prec – the number of terms to be emitted

  • mode – either simple_mode, smallest_mode or teichmuller_mode

EXAMPLES:

sage: E = Zp(5,4)(373).expansion()
sage: I = iter(E)  # indirect doctest
sage: type(I)
<class 'sage.rings.padics.padic_capped_relative_element.ExpansionIter'>

>>> from sage.all import *
>>> E = Zp(Integer(5),Integer(4))(Integer(373)).expansion()
>>> I = iter(E)  # indirect doctest
>>> type(I)
<class 'sage.rings.padics.padic_capped_relative_element.ExpansionIter'>
class sage.rings.padics.padic_capped_absolute_element.ExpansionIterable[source]

Bases: object

An iterable storing a \(p\)-adic expansion of an element.

This class should not be instantiated directly, but instead using expansion().

INPUT:

  • elt – the \(p\)-adic element

  • prec – the number of terms to be emitted

  • val_shift – how many zeros to add at the beginning of the expansion, or the number of initial terms to truncate (if negative)

  • mode – one of the following:

    • 'simple_mode'

    • 'smallest_mode'

    • 'teichmuller_mode'

EXAMPLES:

sage: E = Zp(5,4)(373).expansion()  # indirect doctest
sage: type(E)
<class 'sage.rings.padics.padic_capped_relative_element.ExpansionIterable'>

>>> from sage.all import *
>>> E = Zp(Integer(5),Integer(4))(Integer(373)).expansion()  # indirect doctest
>>> type(E)
<class 'sage.rings.padics.padic_capped_relative_element.ExpansionIterable'>
class sage.rings.padics.padic_capped_absolute_element.PowComputer_[source]

Bases: PowComputer_base

A PowComputer for a capped-absolute \(p\)-adic ring.

sage.rings.padics.padic_capped_absolute_element.make_pAdicCappedAbsoluteElement(parent, x, absprec)[source]

Unpickles a capped absolute element.

EXAMPLES:

sage: from sage.rings.padics.padic_capped_absolute_element import make_pAdicCappedAbsoluteElement
sage: R = ZpCA(5)
sage: a = make_pAdicCappedAbsoluteElement(R, 17*25, 5); a
2*5^2 + 3*5^3 + O(5^5)

>>> from sage.all import *
>>> from sage.rings.padics.padic_capped_absolute_element import make_pAdicCappedAbsoluteElement
>>> R = ZpCA(Integer(5))
>>> a = make_pAdicCappedAbsoluteElement(R, Integer(17)*Integer(25), Integer(5)); a
2*5^2 + 3*5^3 + O(5^5)
class sage.rings.padics.padic_capped_absolute_element.pAdicCappedAbsoluteElement[source]

Bases: CAElement

Construct new element with given parent and value.

INPUT:

  • x – value to coerce into a capped absolute ring

  • absprec – maximum number of digits of absolute precision

  • relprec – maximum number of digits of relative precision

EXAMPLES:

sage: R = ZpCA(3, 5)
sage: R(2)
2 + O(3^5)
sage: R(2, absprec=2)
2 + O(3^2)
sage: R(3, relprec=2)
3 + O(3^3)
sage: R(Qp(3)(10))
1 + 3^2 + O(3^5)
sage: R(pari(6))
2*3 + O(3^5)
sage: R(pari(1/2))
2 + 3 + 3^2 + 3^3 + 3^4 + O(3^5)
sage: R(1/2)
2 + 3 + 3^2 + 3^3 + 3^4 + O(3^5)
sage: R(mod(-1, 3^7))
2 + 2*3 + 2*3^2 + 2*3^3 + 2*3^4 + O(3^5)
sage: R(mod(-1, 3^2))
2 + 2*3 + O(3^2)
sage: R(3 + O(3^2))
3 + O(3^2)

>>> from sage.all import *
>>> R = ZpCA(Integer(3), Integer(5))
>>> R(Integer(2))
2 + O(3^5)
>>> R(Integer(2), absprec=Integer(2))
2 + O(3^2)
>>> R(Integer(3), relprec=Integer(2))
3 + O(3^3)
>>> R(Qp(Integer(3))(Integer(10)))
1 + 3^2 + O(3^5)
>>> R(pari(Integer(6)))
2*3 + O(3^5)
>>> R(pari(Integer(1)/Integer(2)))
2 + 3 + 3^2 + 3^3 + 3^4 + O(3^5)
>>> R(Integer(1)/Integer(2))
2 + 3 + 3^2 + 3^3 + 3^4 + O(3^5)
>>> R(mod(-Integer(1), Integer(3)**Integer(7)))
2 + 2*3 + 2*3^2 + 2*3^3 + 2*3^4 + O(3^5)
>>> R(mod(-Integer(1), Integer(3)**Integer(2)))
2 + 2*3 + O(3^2)
>>> R(Integer(3) + O(Integer(3)**Integer(2)))
3 + O(3^2)
lift()[source]

EXAMPLES:

sage: R = ZpCA(3)
sage: R(10).lift()
10
sage: R(-1).lift()
3486784400

>>> from sage.all import *
>>> R = ZpCA(Integer(3))
>>> R(Integer(10)).lift()
10
>>> R(-Integer(1)).lift()
3486784400
multiplicative_order()[source]

Return the minimum possible multiplicative order of this element.

OUTPUT:

The multiplicative order of self. This is the minimum multiplicative order of all elements of \(\ZZ_p\) lifting self to infinite precision.

EXAMPLES:

sage: R = ZpCA(7, 6)
sage: R(1/3)
5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + 4*7^5 + O(7^6)
sage: R(1/3).multiplicative_order()
+Infinity
sage: R(7).multiplicative_order()
+Infinity
sage: R(1).multiplicative_order()
1
sage: R(-1).multiplicative_order()
2
sage: R.teichmuller(3).multiplicative_order()
6

>>> from sage.all import *
>>> R = ZpCA(Integer(7), Integer(6))
>>> R(Integer(1)/Integer(3))
5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + 4*7^5 + O(7^6)
>>> R(Integer(1)/Integer(3)).multiplicative_order()
+Infinity
>>> R(Integer(7)).multiplicative_order()
+Infinity
>>> R(Integer(1)).multiplicative_order()
1
>>> R(-Integer(1)).multiplicative_order()
2
>>> R.teichmuller(Integer(3)).multiplicative_order()
6
residue(absprec=1, field=None, check_prec=True)[source]

Reduces self modulo \(p^\mathrm{absprec}\).

INPUT:

  • absprec – nonnegative integer (default: 1)

  • field – boolean (default: None); whether to return an element of GF(p) or Zmod(p)

  • check_prec – boolean (default: True); whether to raise an error if this element has insufficient precision to determine the reduction

OUTPUT:

This element reduced modulo \(p^\mathrm{absprec}\) as an element of \(\ZZ/p^\mathrm{absprec}\ZZ\)

EXAMPLES:

sage: R = Zp(7,10,'capped-abs')
sage: a = R(8)
sage: a.residue(1)
1

>>> from sage.all import *
>>> R = Zp(Integer(7),Integer(10),'capped-abs')
>>> a = R(Integer(8))
>>> a.residue(Integer(1))
1

This is different from applying % p^n which returns an element in the same ring:

sage: b = a.residue(2); b
8
sage: b.parent()
Ring of integers modulo 49
sage: c = a % 7^2; c
1 + 7 + O(7^10)
sage: c.parent()
7-adic Ring with capped absolute precision 10

>>> from sage.all import *
>>> b = a.residue(Integer(2)); b
8
>>> b.parent()
Ring of integers modulo 49
>>> c = a % Integer(7)**Integer(2); c
1 + 7 + O(7^10)
>>> c.parent()
7-adic Ring with capped absolute precision 10

Note that reduction of c dropped to the precision of the unit part of 7^2, see _mod_():

sage: R(7^2).unit_part()
1 + O(7^8)

>>> from sage.all import *
>>> R(Integer(7)**Integer(2)).unit_part()
1 + O(7^8)

See also

_mod_()

class sage.rings.padics.padic_capped_absolute_element.pAdicCoercion_CA_frac_field[source]

Bases: RingHomomorphism

The canonical inclusion of Zq into its fraction field.

EXAMPLES:

sage: # needs sage.libs.flint
sage: R.<a> = ZqCA(27, implementation='FLINT')
sage: K = R.fraction_field()
sage: f = K.coerce_map_from(R); f
Ring morphism:
  From: 3-adic Unramified Extension Ring in a defined by x^3 + 2*x + 1
  To:   3-adic Unramified Extension Field in a defined by x^3 + 2*x + 1

>>> from sage.all import *
>>> # needs sage.libs.flint
>>> R = ZqCA(Integer(27), implementation='FLINT', names=('a',)); (a,) = R._first_ngens(1)
>>> K = R.fraction_field()
>>> f = K.coerce_map_from(R); f
Ring morphism:
  From: 3-adic Unramified Extension Ring in a defined by x^3 + 2*x + 1
  To:   3-adic Unramified Extension Field in a defined by x^3 + 2*x + 1
is_injective()[source]

Return whether this map is injective.

EXAMPLES:

sage: # needs sage.libs.flint
sage: R.<a> = ZqCA(9, implementation='FLINT')
sage: K = R.fraction_field()
sage: f = K.coerce_map_from(R)
sage: f.is_injective()
True

>>> from sage.all import *
>>> # needs sage.libs.flint
>>> R = ZqCA(Integer(9), implementation='FLINT', names=('a',)); (a,) = R._first_ngens(1)
>>> K = R.fraction_field()
>>> f = K.coerce_map_from(R)
>>> f.is_injective()
True
is_surjective()[source]

Return whether this map is surjective.

EXAMPLES:

sage: # needs sage.libs.flint
sage: R.<a> = ZqCA(9, implementation='FLINT')
sage: K = R.fraction_field()
sage: f = K.coerce_map_from(R)
sage: f.is_surjective()
False

>>> from sage.all import *
>>> # needs sage.libs.flint
>>> R = ZqCA(Integer(9), implementation='FLINT', names=('a',)); (a,) = R._first_ngens(1)
>>> K = R.fraction_field()
>>> f = K.coerce_map_from(R)
>>> f.is_surjective()
False
section()[source]

Return a map back to the ring that converts elements of nonnegative valuation.

EXAMPLES:

sage: # needs sage.libs.flint
sage: R.<a> = ZqCA(27, implementation='FLINT')
sage: K = R.fraction_field()
sage: f = K.coerce_map_from(R)
sage: f(K.gen())
a + O(3^20)
sage: f.section()
Generic morphism:
  From: 3-adic Unramified Extension Field in a defined by x^3 + 2*x + 1
  To:   3-adic Unramified Extension Ring in a defined by x^3 + 2*x + 1

>>> from sage.all import *
>>> # needs sage.libs.flint
>>> R = ZqCA(Integer(27), implementation='FLINT', names=('a',)); (a,) = R._first_ngens(1)
>>> K = R.fraction_field()
>>> f = K.coerce_map_from(R)
>>> f(K.gen())
a + O(3^20)
>>> f.section()
Generic morphism:
  From: 3-adic Unramified Extension Field in a defined by x^3 + 2*x + 1
  To:   3-adic Unramified Extension Ring in a defined by x^3 + 2*x + 1
class sage.rings.padics.padic_capped_absolute_element.pAdicCoercion_ZZ_CA[source]

Bases: RingHomomorphism

The canonical inclusion from the ring of integers to a capped absolute ring.

EXAMPLES:

sage: f = ZpCA(5).coerce_map_from(ZZ); f
Ring morphism:
  From: Integer Ring
  To:   5-adic Ring with capped absolute precision 20

>>> from sage.all import *
>>> f = ZpCA(Integer(5)).coerce_map_from(ZZ); f
Ring morphism:
  From: Integer Ring
  To:   5-adic Ring with capped absolute precision 20
section()[source]

Return a map back to the ring of integers that approximates an element by an integer.

EXAMPLES:

sage: f = ZpCA(5).coerce_map_from(ZZ).section()
sage: f(ZpCA(5)(-1)) - 5^20
-1

>>> from sage.all import *
>>> f = ZpCA(Integer(5)).coerce_map_from(ZZ).section()
>>> f(ZpCA(Integer(5))(-Integer(1))) - Integer(5)**Integer(20)
-1
class sage.rings.padics.padic_capped_absolute_element.pAdicConvert_CA_ZZ[source]

Bases: RingMap

The map from a capped absolute ring back to the ring of integers that returns the smallest nonnegative integer approximation to its input which is accurate up to the precision.

Raises a ValueError if the input is not in the closure of the image of the ring of integers.

EXAMPLES:

sage: f = ZpCA(5).coerce_map_from(ZZ).section(); f
Set-theoretic ring morphism:
  From: 5-adic Ring with capped absolute precision 20
  To:   Integer Ring

>>> from sage.all import *
>>> f = ZpCA(Integer(5)).coerce_map_from(ZZ).section(); f
Set-theoretic ring morphism:
  From: 5-adic Ring with capped absolute precision 20
  To:   Integer Ring
class sage.rings.padics.padic_capped_absolute_element.pAdicConvert_CA_frac_field[source]

Bases: Morphism

The section of the inclusion from \(\ZZ_q\) to its fraction field.

EXAMPLES:

sage: # needs sage.libs.flint
sage: R.<a> = ZqCA(27, implementation='FLINT')
sage: K = R.fraction_field()
sage: f = R.convert_map_from(K); f
Generic morphism:
  From: 3-adic Unramified Extension Field in a defined by x^3 + 2*x + 1
  To:   3-adic Unramified Extension Ring in a defined by x^3 + 2*x + 1

>>> from sage.all import *
>>> # needs sage.libs.flint
>>> R = ZqCA(Integer(27), implementation='FLINT', names=('a',)); (a,) = R._first_ngens(1)
>>> K = R.fraction_field()
>>> f = R.convert_map_from(K); f
Generic morphism:
  From: 3-adic Unramified Extension Field in a defined by x^3 + 2*x + 1
  To:   3-adic Unramified Extension Ring in a defined by x^3 + 2*x + 1
class sage.rings.padics.padic_capped_absolute_element.pAdicConvert_QQ_CA[source]

Bases: Morphism

The inclusion map from the rationals to a capped absolute ring that is defined on all elements with nonnegative \(p\)-adic valuation.

EXAMPLES:

sage: f = ZpCA(5).convert_map_from(QQ); f
Generic morphism:
  From: Rational Field
  To:   5-adic Ring with capped absolute precision 20

>>> from sage.all import *
>>> f = ZpCA(Integer(5)).convert_map_from(QQ); f
Generic morphism:
  From: Rational Field
  To:   5-adic Ring with capped absolute precision 20
class sage.rings.padics.padic_capped_absolute_element.pAdicTemplateElement[source]

Bases: pAdicGenericElement

A class for common functionality among the \(p\)-adic template classes.

INPUT:

  • parent – a local ring or field

  • x – data defining this element. Various types are supported, including ints, Integers, Rationals, PARI \(p\)-adics, integers mod \(p^k\) and other Sage \(p\)-adics.

  • absprec – a cap on the absolute precision of this element

  • relprec – a cap on the relative precision of this element

EXAMPLES:

sage: Zp(17)(17^3, 8, 4)
17^3 + O(17^7)

>>> from sage.all import *
>>> Zp(Integer(17))(Integer(17)**Integer(3), Integer(8), Integer(4))
17^3 + O(17^7)
expansion(n=None, lift_mode='simple', start_val=None)[source]

Return the coefficients in a \(\pi\)-adic expansion. If this is a field element, start at \(\pi^{\mbox{valuation}}\), if a ring element at \(\pi^0\).

For each lift mode, this function returns a list of \(a_i\) so that this element can be expressed as

\[\pi^v \cdot \sum_{i=0}^\infty a_i \pi^i,\]

where \(v\) is the valuation of this element when the parent is a field, and \(v = 0\) otherwise.

Different lift modes affect the choice of \(a_i\). When lift_mode is 'simple', the resulting \(a_i\) will be nonnegative: if the residue field is \(\GF{p}\) then they will be integers with \(0 \le a_i < p\); otherwise they will be a list of integers in the same range giving the coefficients of a polynomial in the indeterminant representing the maximal unramified subextension.

Choosing lift_mode as 'smallest' is similar to 'simple', but uses a balanced representation \(-p/2 < a_i \le p/2\).

Finally, setting lift_mode = 'teichmuller' will yield Teichmuller representatives for the \(a_i\): \(a_i^q = a_i\). In this case the \(a_i\) will lie in the ring of integers of the maximal unramified subextension of the parent of this element.

INPUT:

  • n – integer (default: None); if given, returns the corresponding entry in the expansion. Can also accept a slice (see slice()).

  • lift_mode – 'simple', 'smallest' or 'teichmuller' (default: 'simple')

  • start_val – start at this valuation rather than the default (\(0\) or the valuation of this element)

OUTPUT:

  • If n is None, an iterable giving a \(\pi\)-adic expansion of this element. For base elements the contents will be integers if lift_mode is 'simple' or 'smallest', and elements of self.parent() if lift_mode is 'teichmuller'.

  • If n is an integer, the coefficient of \(\pi^n\) in the \(\pi\)-adic expansion of this element.

Note

Use slice operators to get a particular range.

EXAMPLES:

sage: R = Zp(7,6); a = R(12837162817); a
3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6)
sage: E = a.expansion(); E
7-adic expansion of 3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6)
sage: list(E)
[3, 4, 4, 0, 4, 0]
sage: sum([c * 7^i for i, c in enumerate(E)]) == a
True
sage: E = a.expansion(lift_mode='smallest'); E
7-adic expansion of 3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6) (balanced)
sage: list(E)
[3, -3, -2, 1, -3, 1]
sage: sum([c * 7^i for i, c in enumerate(E)]) == a
True
sage: E = a.expansion(lift_mode='teichmuller'); E
7-adic expansion of 3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6) (teichmuller)
sage: list(E)
[3 + 4*7 + 6*7^2 + 3*7^3 + 2*7^5 + O(7^6),
0,
5 + 2*7 + 3*7^3 + O(7^4),
1 + O(7^3),
3 + 4*7 + O(7^2),
5 + O(7)]
sage: sum(c * 7^i for i, c in enumerate(E))
3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6)

>>> from sage.all import *
>>> R = Zp(Integer(7),Integer(6)); a = R(Integer(12837162817)); a
3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6)
>>> E = a.expansion(); E
7-adic expansion of 3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6)
>>> list(E)
[3, 4, 4, 0, 4, 0]
>>> sum([c * Integer(7)**i for i, c in enumerate(E)]) == a
True
>>> E = a.expansion(lift_mode='smallest'); E
7-adic expansion of 3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6) (balanced)
>>> list(E)
[3, -3, -2, 1, -3, 1]
>>> sum([c * Integer(7)**i for i, c in enumerate(E)]) == a
True
>>> E = a.expansion(lift_mode='teichmuller'); E
7-adic expansion of 3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6) (teichmuller)
>>> list(E)
[3 + 4*7 + 6*7^2 + 3*7^3 + 2*7^5 + O(7^6),
0,
5 + 2*7 + 3*7^3 + O(7^4),
1 + O(7^3),
3 + 4*7 + O(7^2),
5 + O(7)]
>>> sum(c * Integer(7)**i for i, c in enumerate(E))
3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6)

If the element has positive valuation then the list will start with some zeros:

sage: a = R(7^3 * 17)
sage: E = a.expansion(); E
7-adic expansion of 3*7^3 + 2*7^4 + O(7^9)
sage: list(E)
[0, 0, 0, 3, 2, 0, 0, 0, 0]

>>> from sage.all import *
>>> a = R(Integer(7)**Integer(3) * Integer(17))
>>> E = a.expansion(); E
7-adic expansion of 3*7^3 + 2*7^4 + O(7^9)
>>> list(E)
[0, 0, 0, 3, 2, 0, 0, 0, 0]

The expansion of 0 is truncated:

sage: E = R(0, 7).expansion(); E
7-adic expansion of O(7^7)
sage: len(E)
0
sage: list(E)
[]

>>> from sage.all import *
>>> E = R(Integer(0), Integer(7)).expansion(); E
7-adic expansion of O(7^7)
>>> len(E)
0
>>> list(E)
[]

In fields, on the other hand, the expansion starts at the valuation:

sage: R = Qp(7,4); a = R(6*7+7**2); E = a.expansion(); E
7-adic expansion of 6*7 + 7^2 + O(7^5)
sage: list(E)
[6, 1, 0, 0]
sage: list(a.expansion(lift_mode='smallest'))
[-1, 2, 0, 0]
sage: list(a.expansion(lift_mode='teichmuller'))
[6 + 6*7 + 6*7^2 + 6*7^3 + O(7^4),
2 + 4*7 + 6*7^2 + O(7^3),
3 + 4*7 + O(7^2),
3 + O(7)]

>>> from sage.all import *
>>> R = Qp(Integer(7),Integer(4)); a = R(Integer(6)*Integer(7)+Integer(7)**Integer(2)); E = a.expansion(); E
7-adic expansion of 6*7 + 7^2 + O(7^5)
>>> list(E)
[6, 1, 0, 0]
>>> list(a.expansion(lift_mode='smallest'))
[-1, 2, 0, 0]
>>> list(a.expansion(lift_mode='teichmuller'))
[6 + 6*7 + 6*7^2 + 6*7^3 + O(7^4),
2 + 4*7 + 6*7^2 + O(7^3),
3 + 4*7 + O(7^2),
3 + O(7)]

You can ask for a specific entry in the expansion:

sage: a.expansion(1)
6
sage: a.expansion(1, lift_mode='smallest')
-1
sage: a.expansion(2, lift_mode='teichmuller')
2 + 4*7 + 6*7^2 + O(7^3)

>>> from sage.all import *
>>> a.expansion(Integer(1))
6
>>> a.expansion(Integer(1), lift_mode='smallest')
-1
>>> a.expansion(Integer(2), lift_mode='teichmuller')
2 + 4*7 + 6*7^2 + O(7^3)
lift_to_precision(absprec=None)[source]

Return another element of the same parent with absolute precision at least absprec, congruent to this \(p\)-adic element modulo the precision of this element.

INPUT:

  • absprec – integer or None (default: None); the absolute precision of the result. If None, lifts to the maximum precision allowed.

Note

If setting absprec that high would violate the precision cap, raises a precision error. Note that the new digits will not necessarily be zero.

EXAMPLES:

sage: R = ZpCA(17)
sage: R(-1,2).lift_to_precision(10)
16 + 16*17 + O(17^10)
sage: R(1,15).lift_to_precision(10)
1 + O(17^15)
sage: R(1,15).lift_to_precision(30)
Traceback (most recent call last):
...
PrecisionError: precision higher than allowed by the precision cap
sage: R(-1,2).lift_to_precision().precision_absolute() == R.precision_cap()
True

sage: R = Zp(5); c = R(17,3); c.lift_to_precision(8)
2 + 3*5 + O(5^8)
sage: c.lift_to_precision().precision_relative() == R.precision_cap()
True

>>> from sage.all import *
>>> R = ZpCA(Integer(17))
>>> R(-Integer(1),Integer(2)).lift_to_precision(Integer(10))
16 + 16*17 + O(17^10)
>>> R(Integer(1),Integer(15)).lift_to_precision(Integer(10))
1 + O(17^15)
>>> R(Integer(1),Integer(15)).lift_to_precision(Integer(30))
Traceback (most recent call last):
...
PrecisionError: precision higher than allowed by the precision cap
>>> R(-Integer(1),Integer(2)).lift_to_precision().precision_absolute() == R.precision_cap()
True

>>> R = Zp(Integer(5)); c = R(Integer(17),Integer(3)); c.lift_to_precision(Integer(8))
2 + 3*5 + O(5^8)
>>> c.lift_to_precision().precision_relative() == R.precision_cap()
True

Fixed modulus elements don’t raise errors:

sage: R = ZpFM(5); a = R(5); a.lift_to_precision(7)
5
sage: a.lift_to_precision(10000)
5

>>> from sage.all import *
>>> R = ZpFM(Integer(5)); a = R(Integer(5)); a.lift_to_precision(Integer(7))
5
>>> a.lift_to_precision(Integer(10000))
5
residue(absprec=1, field=None, check_prec=True)[source]

Reduce this element modulo \(p^\mathrm{absprec}\).

INPUT:

  • absprec – 0 or 1

  • field – boolean (default: None); for precision 1, whether to return an element of the residue field or a residue ring. Currently unused.

  • check_prec – boolean (default: True); whether to raise an error if this element has insufficient precision to determine the reduction. Errors are never raised for fixed-mod or floating-point types.

OUTPUT:

This element reduced modulo \(p^\mathrm{absprec}\) as an element of the residue field or the null ring.

EXAMPLES:

sage: # needs sage.libs.ntl
sage: R.<a> = Zq(27, 4)
sage: (3 + 3*a).residue()
0
sage: (a + 1).residue()
a0 + 1

>>> from sage.all import *
>>> # needs sage.libs.ntl
>>> R = Zq(Integer(27), Integer(4), names=('a',)); (a,) = R._first_ngens(1)
>>> (Integer(3) + Integer(3)*a).residue()
0
>>> (a + Integer(1)).residue()
a0 + 1
teichmuller_expansion(n=None)[source]

Return an iterator over coefficients \(a_0, a_1, \dots, a_n\) such that

  • \(a_i^q = a_i\), where \(q\) is the cardinality of the residue field,

  • this element can be expressed as

\[\pi^v \cdot \sum_{i=0}^\infty a_i \pi^i\]

where \(v\) is the valuation of this element when the parent is a field, and \(v = 0\) otherwise.

  • if \(a_i \ne 0\), the precision of \(a_i\) is \(i\) less than the precision of this element (relative in the case that the parent is a field, absolute otherwise)

Note

The coefficients will lie in the ring of integers of the maximal unramified subextension.

INPUT:

  • n – integer (default: None); if given, returns the coefficient of \(\pi^n\) in the expansion

EXAMPLES:

For fields, the expansion starts at the valuation:

sage: R = Qp(5,5); list(R(70).teichmuller_expansion())
[4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + O(5^5),
3 + 3*5 + 2*5^2 + 3*5^3 + O(5^4),
2 + 5 + 2*5^2 + O(5^3),
1 + O(5^2),
4 + O(5)]

>>> from sage.all import *
>>> R = Qp(Integer(5),Integer(5)); list(R(Integer(70)).teichmuller_expansion())
[4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + O(5^5),
3 + 3*5 + 2*5^2 + 3*5^3 + O(5^4),
2 + 5 + 2*5^2 + O(5^3),
1 + O(5^2),
4 + O(5)]

But if you specify n, you get the coefficient of \(\pi^n\):

sage: R(70).teichmuller_expansion(2)
3 + 3*5 + 2*5^2 + 3*5^3 + O(5^4)

>>> from sage.all import *
>>> R(Integer(70)).teichmuller_expansion(Integer(2))
3 + 3*5 + 2*5^2 + 3*5^3 + O(5^4)
unit_part()[source]

Return the unit part of this element.

This is the \(p\)-adic element \(u\) in the same ring so that this element is \(\pi^v u\), where \(\pi\) is a uniformizer and \(v\) is the valuation of this element.

EXAMPLES:

sage: # needs sage.libs.ntl
sage: R.<a> = Zq(125)
sage: (5*a).unit_part()
a + O(5^20)

>>> from sage.all import *
>>> # needs sage.libs.ntl
>>> R = Zq(Integer(125), names=('a',)); (a,) = R._first_ngens(1)
>>> (Integer(5)*a).unit_part()
a + O(5^20)
sage.rings.padics.padic_capped_absolute_element.unpickle_cae_v2(cls, parent, value, absprec)[source]

Unpickle capped absolute elements.

INPUT:

  • cls – the class of the capped absolute element

  • parent – a \(p\)-adic ring

  • value – a Python object wrapping a celement, of the kind accepted by the cunpickle function

  • absprec – a Python int or Sage integer

EXAMPLES:

sage: from sage.rings.padics.padic_capped_absolute_element import unpickle_cae_v2, pAdicCappedAbsoluteElement
sage: R = ZpCA(5,8)
sage: a = unpickle_cae_v2(pAdicCappedAbsoluteElement, R, 42, int(6)); a
2 + 3*5 + 5^2 + O(5^6)
sage: a.parent() is R
True

>>> from sage.all import *
>>> from sage.rings.padics.padic_capped_absolute_element import unpickle_cae_v2, pAdicCappedAbsoluteElement
>>> R = ZpCA(Integer(5),Integer(8))
>>> a = unpickle_cae_v2(pAdicCappedAbsoluteElement, R, Integer(42), int(Integer(6))); a
2 + 3*5 + 5^2 + O(5^6)
>>> a.parent() is R
True