\(p\)-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 elementprec
– the number of terms to be emittedmode
– eithersimple_mode
,smallest_mode
orteichmuller_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 elementprec
– the number of terms to be emittedval_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
– an 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
isNone
, returns ifself == 0
.INPUT:
right
– a \(p\)-adic element with the same parentabsprec
– a positive integer orNone
(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]#
Returns whether
self
is zero modulo \(\pi^{\mbox{absprec}}\).INPUT:
absprec
– an 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 ofself
.If
self == 0
, then the unit part isO(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 padic 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 non-negative 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]#
Returns 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 non-negative 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 non-negative \(p\)-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
– apAdicRingFixedMod
object.x
– input data to be converted into the parent.absprec
– ignored; for compatibility with other \(p\)-adic ringsrelprec
– 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)
fork
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
– an integer (default:1
)field
– boolean (defaultNone
). Whether to return an element of GF(p) or Zmod(p).check_prec
– boolean (defaultFalse
). 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 fieldx
– 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 elementrelprec
– 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 non-negative: 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 (defaultNone
). If given, returns the corresponding entry in the expansion. Can also accept a slice (seeslice()
)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
isNone
, an iterable giving a \(\pi\)-adic expansion of this element. For base elements the contents will be integers iflift_mode
is'simple'
or'smallest'
, and elements ofself.parent()
iflift_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
– an integer orNone
(default:None
); the absolute precision of the result. IfNone
, 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
or1
.field
– boolean (defaultNone
). For precision 1, whether to return an element of the residue field or a residue ring. Currently unused.check_prec
– boolean (defaultTrue
). 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]#
Returns 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 (defaultNone
). 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]#
Returns 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