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Properties

Label 12.0.33096636485...224.15
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 19^{10}$
Root discriminant $288.39$
Ramified primes $2, 3, 7, 19$
Class number $1218048$ (GRH)
Class group $[2, 2, 2, 4, 38064]$ (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![1003941856, -351879680, 227928720, -62065824, 22127456, -3718848, 789272, -77392, 13350, -880, 145, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 145*x^10 - 880*x^9 + 13350*x^8 - 77392*x^7 + 789272*x^6 - 3718848*x^5 + 22127456*x^4 - 62065824*x^3 + 227928720*x^2 - 351879680*x + 1003941856)
 
gp: K = bnfinit(x^12 - 2*x^11 + 145*x^10 - 880*x^9 + 13350*x^8 - 77392*x^7 + 789272*x^6 - 3718848*x^5 + 22127456*x^4 - 62065824*x^3 + 227928720*x^2 - 351879680*x + 1003941856, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 145 x^{10} - 880 x^{9} + 13350 x^{8} - 77392 x^{7} + 789272 x^{6} - 3718848 x^{5} + 22127456 x^{4} - 62065824 x^{3} + 227928720 x^{2} - 351879680 x + 1003941856 \)

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:  \(330966364856493592168450228224=2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $288.39$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3192=2^{3}\cdot 3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{3192}(1,·)$, $\chi_{3192}(2117,·)$, $\chi_{3192}(2273,·)$, $\chi_{3192}(521,·)$, $\chi_{3192}(797,·)$, $\chi_{3192}(1873,·)$, $\chi_{3192}(277,·)$, $\chi_{3192}(677,·)$, $\chi_{3192}(2393,·)$, $\chi_{3192}(121,·)$, $\chi_{3192}(1597,·)$, $\chi_{3192}(1717,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{160} a^{6} + \frac{7}{80} a^{5} + \frac{7}{160} a^{4} + \frac{13}{40} a^{3} - \frac{3}{8} a^{2} - \frac{19}{40}$, $\frac{1}{320} a^{7} + \frac{11}{320} a^{5} + \frac{17}{160} a^{4} + \frac{13}{80} a^{3} - \frac{3}{8} a^{2} - \frac{39}{80} a + \frac{13}{40}$, $\frac{1}{640} a^{8} + \frac{1}{640} a^{6} - \frac{13}{320} a^{5} + \frac{31}{320} a^{4} + \frac{3}{8} a^{3} + \frac{11}{160} a^{2} + \frac{33}{80} a + \frac{7}{16}$, $\frac{1}{1280} a^{9} + \frac{1}{1280} a^{7} + \frac{1}{640} a^{6} + \frac{67}{640} a^{5} - \frac{11}{320} a^{4} + \frac{11}{64} a^{3} + \frac{3}{160} a^{2} + \frac{7}{32} a + \frac{7}{80}$, $\frac{1}{204800} a^{10} + \frac{9}{102400} a^{9} + \frac{87}{204800} a^{8} - \frac{33}{25600} a^{7} - \frac{17}{25600} a^{6} - \frac{91}{1280} a^{5} - \frac{463}{5120} a^{4} - \frac{2479}{6400} a^{3} - \frac{3239}{12800} a^{2} - \frac{85}{256} a - \frac{1713}{12800}$, $\frac{1}{50708785882342182817177600} a^{11} - \frac{10142734291477159573}{50708785882342182817177600} a^{10} - \frac{19787110707280584076711}{50708785882342182817177600} a^{9} + \frac{3465500788208894185079}{50708785882342182817177600} a^{8} - \frac{1047313798414337964817}{3169299117646386426073600} a^{7} - \frac{19682001121111263036013}{6338598235292772852147200} a^{6} - \frac{11815914109167271847695}{253543929411710914085888} a^{5} - \frac{610843031619289006302271}{6338598235292772852147200} a^{4} + \frac{432093852787106534977819}{3169299117646386426073600} a^{3} + \frac{200147988295316210133799}{3169299117646386426073600} a^{2} - \frac{23501686319189537584923}{3169299117646386426073600} a - \frac{1147283342813890690454377}{3169299117646386426073600}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{38064}$, which has order $1218048$ (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:  \( 226303.78002303242 \) (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

\(\Q(\sqrt{-798}) \), \(\Q(\sqrt{-399}) \), \(\Q(\sqrt{2}) \), 3.3.17689.2, \(\Q(\sqrt{2}, \sqrt{-399})\), 6.0.575296762424832.2, 6.0.1123626489111.2, 6.6.160205169152.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$19$19.12.10.3$x^{12} - 19 x^{6} + 5776$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$