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Properties

Label 12.0.21876410091...000.37
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{18}\cdot 5^{6}\cdot 13^{10}$
Root discriminant $278.61$
Ramified primes $2, 3, 5, 13$
Class number $1138176$ (GRH)
Class group $[2, 2, 4, 12, 5928]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5374163484, 279668376, 438168960, 14790204, 18065637, 300456, 470242, 2496, 8703, -52, 102, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 102*x^10 - 52*x^9 + 8703*x^8 + 2496*x^7 + 470242*x^6 + 300456*x^5 + 18065637*x^4 + 14790204*x^3 + 438168960*x^2 + 279668376*x + 5374163484)
 
gp: K = bnfinit(x^12 + 102*x^10 - 52*x^9 + 8703*x^8 + 2496*x^7 + 470242*x^6 + 300456*x^5 + 18065637*x^4 + 14790204*x^3 + 438168960*x^2 + 279668376*x + 5374163484, 1)
 

Normalized defining polynomial

\( x^{12} + 102 x^{10} - 52 x^{9} + 8703 x^{8} + 2496 x^{7} + 470242 x^{6} + 300456 x^{5} + 18065637 x^{4} + 14790204 x^{3} + 438168960 x^{2} + 279668376 x + 5374163484 \)

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:  \(218764100914962817683456000000=2^{18}\cdot 3^{18}\cdot 5^{6}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $278.61$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4680=2^{3}\cdot 3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4680}(1,·)$, $\chi_{4680}(601,·)$, $\chi_{4680}(841,·)$, $\chi_{4680}(989,·)$, $\chi_{4680}(29,·)$, $\chi_{4680}(2669,·)$, $\chi_{4680}(3389,·)$, $\chi_{4680}(121,·)$, $\chi_{4680}(3509,·)$, $\chi_{4680}(2521,·)$, $\chi_{4680}(3481,·)$, $\chi_{4680}(2909,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{12} a^{6} - \frac{1}{6} a^{5} - \frac{1}{4} a^{4} - \frac{1}{6} a^{3} + \frac{1}{12} a^{2} - \frac{1}{2}$, $\frac{1}{12} a^{7} + \frac{1}{12} a^{5} + \frac{1}{3} a^{4} - \frac{1}{4} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{348} a^{8} + \frac{5}{348} a^{7} - \frac{13}{348} a^{6} - \frac{1}{116} a^{5} - \frac{97}{348} a^{4} + \frac{167}{348} a^{3} + \frac{37}{174} a^{2} + \frac{17}{58} a - \frac{14}{29}$, $\frac{1}{1044} a^{9} - \frac{1}{1044} a^{8} - \frac{7}{522} a^{7} - \frac{41}{1044} a^{6} - \frac{83}{522} a^{5} + \frac{169}{1044} a^{4} + \frac{1}{4} a^{3} + \frac{5}{29} a^{2} + \frac{5}{58} a - \frac{1}{29}$, $\frac{1}{11484} a^{10} - \frac{1}{5742} a^{9} - \frac{13}{11484} a^{8} + \frac{13}{638} a^{7} + \frac{223}{11484} a^{6} - \frac{25}{2871} a^{5} + \frac{97}{261} a^{4} + \frac{4}{87} a^{3} + \frac{326}{957} a^{2} + \frac{98}{319} a - \frac{115}{319}$, $\frac{1}{110250300786930775687289834772} a^{11} - \frac{1439533782648300081208669}{110250300786930775687289834772} a^{10} + \frac{9685584687690005117227723}{55125150393465387843644917386} a^{9} - \frac{37847980639446848316773533}{36750100262310258562429944924} a^{8} + \frac{53326298870444768273376397}{9187525065577564640607486231} a^{7} + \frac{80051061496464959353223444}{27562575196732693921822458693} a^{6} + \frac{15266327318746541520100391909}{110250300786930775687289834772} a^{5} + \frac{1364893990392938288183125505}{10022754616993706880662712252} a^{4} - \frac{6435421224096315124423240169}{18375050131155129281214972462} a^{3} + \frac{11981617675972841670005465335}{36750100262310258562429944924} a^{2} + \frac{1494356216638716202335508250}{3062508355192521546869162077} a + \frac{2157232967398685574627854127}{6125016710385043093738324154}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{12}\times C_{5928}$, which has order $1138176$ (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:  \( 16557.868316895623 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-390}) \), \(\Q(\sqrt{13}) \), 3.3.13689.1, \(\Q(\sqrt{13}, \sqrt{-30})\), 6.0.35978634432000.3, 6.0.467722247616000.2, 6.6.2436053373.1

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 R R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$