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Properties

Label 11.11.2786968520...449.10
Degree $11$
Signature $[11, 0]$
Discriminant $11^{20}\cdot 23^{10}$
Root discriminant $1353.24$
Ramified primes $11, 23$
Class number $11$ (GRH)
Class group $[11]$ (GRH)
Galois group $C_{11}$ (as 11T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![152668956353, 103006408615, 11665833289, -3771314393, -853505367, -19353235, 7026316, 461472, -10373, -1265, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 1265*x^9 - 10373*x^8 + 461472*x^7 + 7026316*x^6 - 19353235*x^5 - 853505367*x^4 - 3771314393*x^3 + 11665833289*x^2 + 103006408615*x + 152668956353)
 
gp: K = bnfinit(x^11 - 1265*x^9 - 10373*x^8 + 461472*x^7 + 7026316*x^6 - 19353235*x^5 - 853505367*x^4 - 3771314393*x^3 + 11665833289*x^2 + 103006408615*x + 152668956353, 1)
 

Normalized defining polynomial

\( x^{11} - 1265 x^{9} - 10373 x^{8} + 461472 x^{7} + 7026316 x^{6} - 19353235 x^{5} - 853505367 x^{4} - 3771314393 x^{3} + 11665833289 x^{2} + 103006408615 x + 152668956353 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $11$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(27869685209056005146671841116784449=11^{20}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1353.24$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2783=11^{2}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{2783}(2465,·)$, $\chi_{2783}(133,·)$, $\chi_{2783}(1,·)$, $\chi_{2783}(936,·)$, $\chi_{2783}(1002,·)$, $\chi_{2783}(2124,·)$, $\chi_{2783}(1409,·)$, $\chi_{2783}(2036,·)$, $\chi_{2783}(837,·)$, $\chi_{2783}(2234,·)$, $\chi_{2783}(991,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{643873098778057590850178338274046335308081} a^{10} - \frac{283191993144505533565795257823515501536771}{643873098778057590850178338274046335308081} a^{9} - \frac{195483673600319742998243996887126869989798}{643873098778057590850178338274046335308081} a^{8} - \frac{229718895717206571480077698266460330181609}{643873098778057590850178338274046335308081} a^{7} + \frac{31929819680699788417326643496938822212180}{643873098778057590850178338274046335308081} a^{6} - \frac{196587157576367621235046544554850365502118}{643873098778057590850178338274046335308081} a^{5} - \frac{375968459195859371955072453301569961374}{643873098778057590850178338274046335308081} a^{4} - \frac{297563517420679426234332622672742118960469}{643873098778057590850178338274046335308081} a^{3} + \frac{247442423397740609445510228698568643612788}{643873098778057590850178338274046335308081} a^{2} + \frac{238367505274388557045920022858283115030429}{643873098778057590850178338274046335308081} a - \frac{198438015724935414960691254769883315612615}{643873098778057590850178338274046335308081}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{11}$, which has order $11$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2083946923676.7688 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{11}$ (as 11T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 11
The 11 conjugacy class representatives for $C_{11}$
Character table for $C_{11}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ R ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ R ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$11$11.11.20.1$x^{11} - 11 x^{10} + 374$$11$$1$$20$$C_{11}$$[2]$
$23$23.11.10.5$x^{11} + 184$$11$$1$$10$$C_{11}$$[\ ]_{11}$