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Properties

Label 11.11.1578628494...3801.1
Degree $11$
Signature $[11, 0]$
Discriminant $331^{10}$
Root discriminant $195.32$
Ramified prime $331$
Class number $1$ (GRH)
Class group Trivial (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![-2287616, 4015168, -1913488, -343344, 475464, -62124, -28617, 6577, 402, -150, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 150*x^9 + 402*x^8 + 6577*x^7 - 28617*x^6 - 62124*x^5 + 475464*x^4 - 343344*x^3 - 1913488*x^2 + 4015168*x - 2287616)
 
gp: K = bnfinit(x^11 - x^10 - 150*x^9 + 402*x^8 + 6577*x^7 - 28617*x^6 - 62124*x^5 + 475464*x^4 - 343344*x^3 - 1913488*x^2 + 4015168*x - 2287616, 1)
 

Normalized defining polynomial

\( x^{11} - x^{10} - 150 x^{9} + 402 x^{8} + 6577 x^{7} - 28617 x^{6} - 62124 x^{5} + 475464 x^{4} - 343344 x^{3} - 1913488 x^{2} + 4015168 x - 2287616 \)

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:  \(15786284949774657045043801=331^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $195.32$
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:  $331$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(331\)
Dirichlet character group:    $\lbrace$$\chi_{331}(1,·)$, $\chi_{331}(293,·)$, $\chi_{331}(167,·)$, $\chi_{331}(74,·)$, $\chi_{331}(270,·)$, $\chi_{331}(111,·)$, $\chi_{331}(80,·)$, $\chi_{331}(274,·)$, $\chi_{331}(180,·)$, $\chi_{331}(85,·)$, $\chi_{331}(120,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{32} a^{7} - \frac{1}{16} a^{4} + \frac{7}{32} a^{3} + \frac{1}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{512} a^{8} - \frac{1}{64} a^{7} - \frac{3}{256} a^{6} + \frac{1}{512} a^{4} + \frac{3}{64} a^{3} + \frac{1}{128} a^{2} + \frac{7}{32} a$, $\frac{1}{84992} a^{9} + \frac{17}{21248} a^{8} + \frac{61}{42496} a^{7} + \frac{11}{10624} a^{6} - \frac{3359}{84992} a^{5} + \frac{177}{21248} a^{4} - \frac{4711}{21248} a^{3} + \frac{425}{2656} a^{2} + \frac{17}{1328} a + \frac{28}{83}$, $\frac{1}{4274757632} a^{10} - \frac{16987}{4274757632} a^{9} + \frac{395191}{534344704} a^{8} - \frac{26290239}{2137378816} a^{7} + \frac{24525597}{4274757632} a^{6} - \frac{236300475}{4274757632} a^{5} + \frac{22458545}{2137378816} a^{4} - \frac{233792547}{1068689408} a^{3} - \frac{69172087}{534344704} a^{2} - \frac{59016867}{133586176} a - \frac{1449553}{4174568}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 1340143683620 \) (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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

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
331Data not computed