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Properties

Label 12.0.19366119397...496.10
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 13^{10}\cdot 19^{6}$
Root discriminant $128.01$
Ramified primes $2, 3, 13, 19$
Class number $325872$ (GRH)
Class group $[2, 2, 81468]$ (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![41405864761, -1357569336, 3992362134, -110101244, 163992613, -3667870, 3673382, -62758, 47339, -552, 333, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 333*x^10 - 552*x^9 + 47339*x^8 - 62758*x^7 + 3673382*x^6 - 3667870*x^5 + 163992613*x^4 - 110101244*x^3 + 3992362134*x^2 - 1357569336*x + 41405864761)
 
gp: K = bnfinit(x^12 - 2*x^11 + 333*x^10 - 552*x^9 + 47339*x^8 - 62758*x^7 + 3673382*x^6 - 3667870*x^5 + 163992613*x^4 - 110101244*x^3 + 3992362134*x^2 - 1357569336*x + 41405864761, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 333 x^{10} - 552 x^{9} + 47339 x^{8} - 62758 x^{7} + 3673382 x^{6} - 3667870 x^{5} + 163992613 x^{4} - 110101244 x^{3} + 3992362134 x^{2} - 1357569336 x + 41405864761 \)

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:  \(19366119397482188691050496=2^{12}\cdot 3^{6}\cdot 13^{10}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $128.01$
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:  $2, 3, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2964=2^{2}\cdot 3\cdot 13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2964}(1,·)$, $\chi_{2964}(2051,·)$, $\chi_{2964}(2053,·)$, $\chi_{2964}(2279,·)$, $\chi_{2964}(685,·)$, $\chi_{2964}(911,·)$, $\chi_{2964}(913,·)$, $\chi_{2964}(2963,·)$, $\chi_{2964}(1141,·)$, $\chi_{2964}(1369,·)$, $\chi_{2964}(1595,·)$, $\chi_{2964}(1823,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{309508396582036859462781284595630565} a^{11} - \frac{39105611850034646706614929887119468}{309508396582036859462781284595630565} a^{10} + \frac{154692838232215243608351217410177096}{309508396582036859462781284595630565} a^{9} - \frac{149803967408543409095498931609767093}{309508396582036859462781284595630565} a^{8} + \frac{90533306381647811679042254112459042}{309508396582036859462781284595630565} a^{7} - \frac{6447897576754576434412595954018856}{61901679316407371892556256919126113} a^{6} + \frac{32452313877013765742742060371488747}{309508396582036859462781284595630565} a^{5} + \frac{128436804997761976002739082923637538}{309508396582036859462781284595630565} a^{4} - \frac{10421833542643986090967284355744529}{61901679316407371892556256919126113} a^{3} - \frac{106269377846458113030806948882725779}{309508396582036859462781284595630565} a^{2} - \frac{95164883665010117714685993854698177}{309508396582036859462781284595630565} a - \frac{149282462514666260434617973309042954}{309508396582036859462781284595630565}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{81468}$, which has order $325872$ (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:  \( 120.78403136265631 \) (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

\(\Q(\sqrt{-741}) \), \(\Q(\sqrt{-57}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\sqrt{13}, \sqrt{-57})\), 6.0.4400695331136.3, 6.0.338515025472.3, \(\Q(\zeta_{13})^+\)

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 R R ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$19$19.12.6.1$x^{12} + 41154 x^{6} - 2476099 x^{2} + 423412929$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$