\(p\)-adic Fixed-Mod Element

Elements of \(p\)-adic Rings with Fixed Modulus

AUTHORS:

  • David Roe

  • Genya Zaytman: documentation

  • David Harvey: doctests

class sage.rings.padics.padic_fixed_mod_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_fixed_mod_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_fixed_mod_element.FMElement[source]

Bases: pAdicTemplateElement

add_bigoh(absprec)[source]

Return a new element truncated modulo \(\pi^{\mbox{absprec}}\).

INPUT:

  • absprec – integer or infinity

OUTPUT: a new element truncated modulo \(\pi^{\mbox{absprec}}\)

EXAMPLES:

sage: R = Zp(7,4,'fixed-mod','series'); a = R(8); a.add_bigoh(1)
1

>>> from sage.all import *
>>> R = Zp(Integer(7),Integer(4),'fixed-mod','series'); a = R(Integer(8)); a.add_bigoh(Integer(1))
1
is_equal_to(_right, absprec=None)[source]

Return whether this element is equal to right modulo \(p^{\mbox{absprec}}\).

If absprec is None, returns if self == 0.

INPUT:

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

  • absprec – positive integer or None (default: None)

EXAMPLES:

sage: R = ZpFM(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

>>> from sage.all import *
>>> R = ZpFM(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
is_zero(absprec=None)[source]

Return whether self is zero modulo \(\pi^{\mbox{absprec}}\).

INPUT:

  • absprec – integer

EXAMPLES:

sage: R = ZpFM(17, 6)
sage: R(0).is_zero()
True
sage: R(17^6).is_zero()
True
sage: R(17^2).is_zero(absprec=2)
True

>>> from sage.all import *
>>> R = ZpFM(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
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> = ZqFM(5^3)
sage: a.polynomial()
x
sage: a.polynomial(var='y')
y
sage: (5*a^2 + 25).polynomial()
5*x^2 + 5^2

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

The absolute precision of this element.

EXAMPLES:

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

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

The relative precision of this element.

EXAMPLES:

sage: R = Zp(7,4,'fixed-mod'); a = R(7); a.precision_relative()
3
sage: a = R(0); a.precision_relative()
0

>>> from sage.all import *
>>> R = Zp(Integer(7),Integer(4),'fixed-mod'); a = R(Integer(7)); a.precision_relative()
3
>>> a = R(Integer(0)); a.precision_relative()
0
unit_part()[source]

Return the unit part of self.

If the valuation of self is positive, then the high digits of the result will be zero.

EXAMPLES:

sage: R = Zp(17, 4, 'fixed-mod')
sage: R(5).unit_part()
5
sage: R(18*17).unit_part()
1 + 17
sage: R(0).unit_part()
0
sage: type(R(5).unit_part())
<class 'sage.rings.padics.padic_fixed_mod_element.pAdicFixedModElement'>
sage: R = ZpFM(5, 5); a = R(75); a.unit_part()
3

>>> from sage.all import *
>>> R = Zp(Integer(17), Integer(4), 'fixed-mod')
>>> R(Integer(5)).unit_part()
5
>>> R(Integer(18)*Integer(17)).unit_part()
1 + 17
>>> R(Integer(0)).unit_part()
0
>>> type(R(Integer(5)).unit_part())
<class 'sage.rings.padics.padic_fixed_mod_element.pAdicFixedModElement'>
>>> R = ZpFM(Integer(5), Integer(5)); a = R(Integer(75)); a.unit_part()
3
val_unit()[source]

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

If self == 0, then the unit part is O(p^self.parent().precision_cap()).

EXAMPLES:

sage: R = ZpFM(5,5)
sage: a = R(75); b = a - a
sage: a.val_unit()
(2, 3)
sage: b.val_unit()
(5, 0)

>>> from sage.all import *
>>> R = ZpFM(Integer(5),Integer(5))
>>> a = R(Integer(75)); b = a - a
>>> a.val_unit()
(2, 3)
>>> b.val_unit()
(5, 0)
class sage.rings.padics.padic_fixed_mod_element.PowComputer_[source]

Bases: PowComputer_base

A PowComputer for a fixed-modulus \(p\)-adic ring.

sage.rings.padics.padic_fixed_mod_element.make_pAdicFixedModElement(parent, value)[source]

Unpickles a fixed modulus element.

EXAMPLES:

sage: from sage.rings.padics.padic_fixed_mod_element import make_pAdicFixedModElement
sage: R = ZpFM(5)
sage: a = make_pAdicFixedModElement(R, 17*25); a
2*5^2 + 3*5^3

>>> from sage.all import *
>>> from sage.rings.padics.padic_fixed_mod_element import make_pAdicFixedModElement
>>> R = ZpFM(Integer(5))
>>> a = make_pAdicFixedModElement(R, Integer(17)*Integer(25)); a
2*5^2 + 3*5^3
class sage.rings.padics.padic_fixed_mod_element.pAdicCoercion_FM_frac_field[source]

Bases: RingHomomorphism

The canonical inclusion of \(\ZZ_q\) into its fraction field.

EXAMPLES:

sage: # needs sage.libs.flint
sage: R.<a> = ZqFM(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 = ZqFM(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> = ZqFM(9)
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 = ZqFM(Integer(9), 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> = ZqFM(9)
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 = ZqFM(Integer(9), 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> = ZqFM(27)
sage: K = R.fraction_field()
sage: f = K.coerce_map_from(R)
sage: f.section()(K.gen())
a

>>> from sage.all import *
>>> # needs sage.libs.flint
>>> R = ZqFM(Integer(27), names=('a',)); (a,) = R._first_ngens(1)
>>> K = R.fraction_field()
>>> f = K.coerce_map_from(R)
>>> f.section()(K.gen())
a
class sage.rings.padics.padic_fixed_mod_element.pAdicCoercion_ZZ_FM[source]

Bases: RingHomomorphism

The canonical inclusion from \(\ZZ\) to a fixed modulus ring.

EXAMPLES:

sage: f = ZpFM(5).coerce_map_from(ZZ); f
Ring morphism:
  From: Integer Ring
  To:   5-adic Ring of fixed modulus 5^20

>>> from sage.all import *
>>> f = ZpFM(Integer(5)).coerce_map_from(ZZ); f
Ring morphism:
  From: Integer Ring
  To:   5-adic Ring of fixed modulus 5^20
section()[source]

Return a map back to \(\ZZ\) that approximates an element of this \(p\)-adic ring by an integer.

EXAMPLES:

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

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

Bases: RingMap

The map from a fixed modulus ring back to \(\ZZ\) that returns the smallest nonnegative integer approximation to its input which is accurate up to the precision.

If the input is not in the closure of the image of \(\ZZ\), raises a ValueError.

EXAMPLES:

sage: f = ZpFM(5).coerce_map_from(ZZ).section(); f
Set-theoretic ring morphism:
  From: 5-adic Ring of fixed modulus 5^20
  To:   Integer Ring

>>> from sage.all import *
>>> f = ZpFM(Integer(5)).coerce_map_from(ZZ).section(); f
Set-theoretic ring morphism:
  From: 5-adic Ring of fixed modulus 5^20
  To:   Integer Ring
class sage.rings.padics.padic_fixed_mod_element.pAdicConvert_FM_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> = ZqFM(27)
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 = ZqFM(Integer(27), 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_fixed_mod_element.pAdicConvert_QQ_FM[source]

Bases: Morphism

The inclusion map from \(\QQ\) to a fixed modulus ring that is defined on all elements with nonnegative \(p\)-adic valuation.

EXAMPLES:

sage: f = ZpFM(5).convert_map_from(QQ); f
Generic morphism:
  From: Rational Field
  To:   5-adic Ring of fixed modulus 5^20

>>> from sage.all import *
>>> f = ZpFM(Integer(5)).convert_map_from(QQ); f
Generic morphism:
  From: Rational Field
  To:   5-adic Ring of fixed modulus 5^20
class sage.rings.padics.padic_fixed_mod_element.pAdicFixedModElement[source]

Bases: FMElement

INPUT:

  • parent – a pAdicRingFixedMod object

  • x – input data to be converted into the parent

  • absprec – ignored; for compatibility with other \(p\)-adic rings

  • relprec – ignored; for compatibility with other \(p\)-adic rings

Note

The following types are currently supported for x:

  • Integers

  • Rationals – denominator must be relatively prime to \(p\)

  • FixedMod \(p\)-adics

  • Elements of IntegerModRing(p^k) for k less than or equal to the modulus

The following types should be supported eventually:

  • Finite precision \(p\)-adics

  • Lazy \(p\)-adics

  • Elements of local extensions of THIS \(p\)-adic ring that actually lie in \(\ZZ_p\)

EXAMPLES:

sage: R = Zp(5, 20, 'fixed-mod', 'terse')

>>> from sage.all import *
>>> R = Zp(Integer(5), Integer(20), 'fixed-mod', 'terse')

Construct from integers:

sage: R(3)
3
sage: R(75)
75
sage: R(0)
0

sage: R(-1)
95367431640624
sage: R(-5)
95367431640620

>>> from sage.all import *
>>> R(Integer(3))
3
>>> R(Integer(75))
75
>>> R(Integer(0))
0

>>> R(-Integer(1))
95367431640624
>>> R(-Integer(5))
95367431640620

Construct from rationals:

sage: R(1/2)
47683715820313
sage: R(-7875/874)
9493096742250
sage: R(15/425)
Traceback (most recent call last):
...
ValueError: p divides denominator

>>> from sage.all import *
>>> R(Integer(1)/Integer(2))
47683715820313
>>> R(-Integer(7875)/Integer(874))
9493096742250
>>> R(Integer(15)/Integer(425))
Traceback (most recent call last):
...
ValueError: p divides denominator

Construct from IntegerMod:

sage: R(Integers(125)(3))
3
sage: R(Integers(5)(3))
3
sage: R(Integers(5^30)(3))
3
sage: R(Integers(5^30)(1+5^23))
1
sage: R(Integers(49)(3))
Traceback (most recent call last):
...
TypeError: p does not divide modulus 49

sage: R(Integers(48)(3))
Traceback (most recent call last):
...
TypeError: p does not divide modulus 48

>>> from sage.all import *
>>> R(Integers(Integer(125))(Integer(3)))
3
>>> R(Integers(Integer(5))(Integer(3)))
3
>>> R(Integers(Integer(5)**Integer(30))(Integer(3)))
3
>>> R(Integers(Integer(5)**Integer(30))(Integer(1)+Integer(5)**Integer(23)))
1
>>> R(Integers(Integer(49))(Integer(3)))
Traceback (most recent call last):
...
TypeError: p does not divide modulus 49

>>> R(Integers(Integer(48))(Integer(3)))
Traceback (most recent call last):
...
TypeError: p does not divide modulus 48

Some other conversions:

sage: R(R(5))
5

>>> from sage.all import *
>>> R(R(Integer(5)))
5

Todo

doctests for converting from other types of \(p\)-adic rings

lift()[source]

Return an integer congruent to self modulo the precision.

Warning

Since fixed modulus elements don’t track their precision, the result may not be correct modulo \(i^{\mathrm{prec_cap}}\) if the element was defined by constructions that lost precision.

EXAMPLES:

sage: R = Zp(7,4,'fixed-mod'); a = R(8); a.lift()
8
sage: type(a.lift())
<class 'sage.rings.integer.Integer'>

>>> from sage.all import *
>>> R = Zp(Integer(7),Integer(4),'fixed-mod'); a = R(Integer(8)); a.lift()
8
>>> type(a.lift())
<class 'sage.rings.integer.Integer'>
multiplicative_order()[source]

Return the minimum possible multiplicative order of self.

OUTPUT:

an integer – the multiplicative order of this element. This is the minimum multiplicative order of all elements of \(\ZZ_p\) lifting this element to infinite precision.

EXAMPLES:

sage: R = ZpFM(7, 6)
sage: R(1/3)
5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + 4*7^5
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 = ZpFM(Integer(7), Integer(6))
>>> R(Integer(1)/Integer(3))
5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + 4*7^5
>>> 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=False)[source]

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

INPUT:

  • absprec – integer (default: 1)

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

  • check_prec – boolean (default: False); no effect (for compatibility with other types)

OUTPUT:

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

EXAMPLES:

sage: R = Zp(7,4,'fixed-mod')
sage: a = R(8)
sage: a.residue(1)
1

>>> from sage.all import *
>>> R = Zp(Integer(7),Integer(4),'fixed-mod')
>>> 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
sage: c.parent()
7-adic Ring of fixed modulus 7^4

>>> 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
>>> c.parent()
7-adic Ring of fixed modulus 7^4

See also

_mod_()

class sage.rings.padics.padic_fixed_mod_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_fixed_mod_element.unpickle_fme_v2(cls, parent, value)[source]

Unpickles a fixed-mod element.

EXAMPLES:

sage: from sage.rings.padics.padic_fixed_mod_element import pAdicFixedModElement, unpickle_fme_v2
sage: R = ZpFM(5)
sage: a = unpickle_fme_v2(pAdicFixedModElement, R, 17*25); a
2*5^2 + 3*5^3
sage: a.parent() is R
True

>>> from sage.all import *
>>> from sage.rings.padics.padic_fixed_mod_element import pAdicFixedModElement, unpickle_fme_v2
>>> R = ZpFM(Integer(5))
>>> a = unpickle_fme_v2(pAdicFixedModElement, R, Integer(17)*Integer(25)); a
2*5^2 + 3*5^3
>>> a.parent() is R
True