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Properties

Label 12.0.37311914733...2533.1
Degree $12$
Signature $[0, 6]$
Discriminant $13^{11}\cdot 113^{6}$
Root discriminant $111.60$
Ramified primes $13, 113$
Class number $102280$ (GRH)
Class group $[2, 51140]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7946060669, -1681486717, 1681486717, -115343229, 115343229, -3475837, 3475837, -51325, 51325, -365, 365, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 365*x^10 - 365*x^9 + 51325*x^8 - 51325*x^7 + 3475837*x^6 - 3475837*x^5 + 115343229*x^4 - 115343229*x^3 + 1681486717*x^2 - 1681486717*x + 7946060669)
 
gp: K = bnfinit(x^12 - x^11 + 365*x^10 - 365*x^9 + 51325*x^8 - 51325*x^7 + 3475837*x^6 - 3475837*x^5 + 115343229*x^4 - 115343229*x^3 + 1681486717*x^2 - 1681486717*x + 7946060669, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 365 x^{10} - 365 x^{9} + 51325 x^{8} - 51325 x^{7} + 3475837 x^{6} - 3475837 x^{5} + 115343229 x^{4} - 115343229 x^{3} + 1681486717 x^{2} - 1681486717 x + 7946060669 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3731191473321768182792533=13^{11}\cdot 113^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $111.60$
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:  $13, 113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1469=13\cdot 113\)
Dirichlet character group:    $\lbrace$$\chi_{1469}(1,·)$, $\chi_{1469}(903,·)$, $\chi_{1469}(1129,·)$, $\chi_{1469}(679,·)$, $\chi_{1469}(112,·)$, $\chi_{1469}(792,·)$, $\chi_{1469}(114,·)$, $\chi_{1469}(564,·)$, $\chi_{1469}(1016,·)$, $\chi_{1469}(1018,·)$, $\chi_{1469}(1244,·)$, $\chi_{1469}(1242,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{888213509} a^{7} - \frac{438521650}{888213509} a^{6} + \frac{196}{888213509} a^{5} + \frac{50084047}{888213509} a^{4} + \frac{10976}{888213509} a^{3} + \frac{327102956}{888213509} a^{2} + \frac{153664}{888213509} a + \frac{61499484}{888213509}$, $\frac{1}{888213509} a^{8} + \frac{224}{888213509} a^{6} - \frac{156382926}{888213509} a^{5} + \frac{15680}{888213509} a^{4} + \frac{311728085}{888213509} a^{3} + \frac{351232}{888213509} a^{2} - \frac{153748710}{888213509} a + \frac{1229312}{888213509}$, $\frac{1}{888213509} a^{9} + \frac{368980684}{888213509} a^{6} - \frac{28224}{888213509} a^{5} - \frac{248536335}{888213509} a^{4} - \frac{2107392}{888213509} a^{3} + \frac{296910393}{888213509} a^{2} - \frac{33191424}{888213509} a + \frac{435531728}{888213509}$, $\frac{1}{888213509} a^{10} - \frac{35280}{888213509} a^{6} + \frac{264757339}{888213509} a^{5} - \frac{3292800}{888213509} a^{4} - \frac{269689660}{888213509} a^{3} - \frac{82978560}{888213509} a^{2} - \frac{391160942}{888213509} a - \frac{309786624}{888213509}$, $\frac{1}{888213509} a^{11} + \frac{123845101}{888213509} a^{6} + \frac{3622080}{888213509} a^{5} + \frac{38819099}{888213509} a^{4} + \frac{304254720}{888213509} a^{3} + \frac{131217810}{888213509} a^{2} - \frac{217801758}{888213509} a - \frac{203806967}{888213509}$

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

Class group and class number

$C_{2}\times C_{51140}$, which has order $102280$ (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:  $5$
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:  \( 120.784031363 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 4.0.28053493.1, \(\Q(\zeta_{13})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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
$13$13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$113$113.6.3.1$x^{6} - 226 x^{4} + 12769 x^{2} - 36072425$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
113.6.3.1$x^{6} - 226 x^{4} + 12769 x^{2} - 36072425$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$