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Properties

Label 11.11.1667785638...3801.1
Degree $11$
Signature $[11, 0]$
Discriminant $419^{10}$
Root discriminant $242.01$
Ramified prime $419$
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![-785489, 873500, 1263744, -2399676, 1224028, -141765, -51221, 10985, 547, -190, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 190*x^9 + 547*x^8 + 10985*x^7 - 51221*x^6 - 141765*x^5 + 1224028*x^4 - 2399676*x^3 + 1263744*x^2 + 873500*x - 785489)
 
gp: K = bnfinit(x^11 - x^10 - 190*x^9 + 547*x^8 + 10985*x^7 - 51221*x^6 - 141765*x^5 + 1224028*x^4 - 2399676*x^3 + 1263744*x^2 + 873500*x - 785489, 1)
 

Normalized defining polynomial

\( x^{11} - x^{10} - 190 x^{9} + 547 x^{8} + 10985 x^{7} - 51221 x^{6} - 141765 x^{5} + 1224028 x^{4} - 2399676 x^{3} + 1263744 x^{2} + 873500 x - 785489 \)

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:  \(166778563814477267272573801=419^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $242.01$
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:  $419$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(419\)
Dirichlet character group:    $\lbrace$$\chi_{419}(129,·)$, $\chi_{419}(69,·)$, $\chi_{419}(102,·)$, $\chi_{419}(1,·)$, $\chi_{419}(169,·)$, $\chi_{419}(300,·)$, $\chi_{419}(13,·)$, $\chi_{419}(334,·)$, $\chi_{419}(152,·)$, $\chi_{419}(59,·)$, $\chi_{419}(348,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{1}{7}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{5537} a^{9} - \frac{246}{5537} a^{8} + \frac{241}{5537} a^{7} - \frac{80}{5537} a^{6} + \frac{1}{113} a^{5} + \frac{17}{113} a^{4} - \frac{57}{5537} a^{3} + \frac{1429}{5537} a^{2} + \frac{2139}{5537} a + \frac{1228}{5537}$, $\frac{1}{6935321084327411153} a^{10} + \frac{28135467939379}{6935321084327411153} a^{9} - \frac{468466111472408428}{6935321084327411153} a^{8} - \frac{1679438060851387}{34850859720238247} a^{7} + \frac{438970130273034}{14478749654128207} a^{6} - \frac{43345154487319583}{141537164986273697} a^{5} + \frac{20102708070478772}{77924956003678777} a^{4} + \frac{1018537295517919104}{6935321084327411153} a^{3} - \frac{1456307496345536}{14478749654128207} a^{2} + \frac{1982582314766429689}{6935321084327411153} a - \frac{1060606824264002241}{6935321084327411153}$

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:  \( 13726169613.8 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/11.11.0.1}{11} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

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
419Data not computed