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Properties

Label 11.11.5800958920...3049.1
Degree $11$
Signature $[11, 0]$
Discriminant $947^{10}$
Root discriminant $507.89$
Ramified prime $947$
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![-47440673, -30077229, 20981317, 8674010, -3028650, -760875, 141142, 28780, -2035, -430, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 430*x^9 - 2035*x^8 + 28780*x^7 + 141142*x^6 - 760875*x^5 - 3028650*x^4 + 8674010*x^3 + 20981317*x^2 - 30077229*x - 47440673)
 
gp: K = bnfinit(x^11 - x^10 - 430*x^9 - 2035*x^8 + 28780*x^7 + 141142*x^6 - 760875*x^5 - 3028650*x^4 + 8674010*x^3 + 20981317*x^2 - 30077229*x - 47440673, 1)
 

Normalized defining polynomial

\( x^{11} - x^{10} - 430 x^{9} - 2035 x^{8} + 28780 x^{7} + 141142 x^{6} - 760875 x^{5} - 3028650 x^{4} + 8674010 x^{3} + 20981317 x^{2} - 30077229 x - 47440673 \)

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:  \(580095892065127629623589183049=947^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $507.89$
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:  $947$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(947\)
Dirichlet character group:    $\lbrace$$\chi_{947}(1,·)$, $\chi_{947}(930,·)$, $\chi_{947}(643,·)$, $\chi_{947}(580,·)$, $\chi_{947}(133,·)$, $\chi_{947}(289,·)$, $\chi_{947}(557,·)$, $\chi_{947}(433,·)$, $\chi_{947}(215,·)$, $\chi_{947}(185,·)$, $\chi_{947}(769,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{2303} a^{8} + \frac{32}{2303} a^{7} - \frac{64}{2303} a^{6} - \frac{1048}{2303} a^{5} - \frac{1117}{2303} a^{4} - \frac{1035}{2303} a^{3} - \frac{1088}{2303} a^{2} + \frac{36}{329} a + \frac{9}{47}$, $\frac{1}{12972799} a^{9} + \frac{955}{12972799} a^{8} + \frac{14205}{301693} a^{7} - \frac{307199}{12972799} a^{6} + \frac{359423}{12972799} a^{5} + \frac{102789}{276017} a^{4} + \frac{4732675}{12972799} a^{3} + \frac{2642664}{12972799} a^{2} + \frac{778382}{1853257} a + \frac{127358}{264751}$, $\frac{1}{14295205466575051319} a^{10} - \frac{483063088171}{14295205466575051319} a^{9} - \frac{2543216346299542}{14295205466575051319} a^{8} - \frac{379762534711941963}{14295205466575051319} a^{7} + \frac{752059469689337147}{14295205466575051319} a^{6} - \frac{4185529375121989192}{14295205466575051319} a^{5} - \frac{55701689073516993}{291738887072960231} a^{4} - \frac{3173116368964403704}{14295205466575051319} a^{3} - \frac{3889484805795192778}{14295205466575051319} a^{2} - \frac{570166728038867870}{2042172209510721617} a + \frac{136314565007939882}{291738887072960231}$

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:  \( 1104123814020 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }$ ${\href{/LocalNumberField/5.11.0.1}{11} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/11.11.0.1}{11} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }$

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
947Data not computed