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Properties

Label 12.0.17099484234...0625.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{18}\cdot 5^{6}\cdot 7^{10}$
Root discriminant $58.81$
Ramified primes $3, 5, 7$
Class number $1554$ (GRH)
Class group $[1554]$ (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![50176, -23520, 209916, 26264, -1911, -3528, 3836, 840, 126, -56, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 56*x^9 + 126*x^8 + 840*x^7 + 3836*x^6 - 3528*x^5 - 1911*x^4 + 26264*x^3 + 209916*x^2 - 23520*x + 50176)
 
gp: K = bnfinit(x^12 - 56*x^9 + 126*x^8 + 840*x^7 + 3836*x^6 - 3528*x^5 - 1911*x^4 + 26264*x^3 + 209916*x^2 - 23520*x + 50176, 1)
 

Normalized defining polynomial

\( x^{12} - 56 x^{9} + 126 x^{8} + 840 x^{7} + 3836 x^{6} - 3528 x^{5} - 1911 x^{4} + 26264 x^{3} + 209916 x^{2} - 23520 x + 50176 \)

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:  \(1709948423405886890625=3^{18}\cdot 5^{6}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.81$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(1,·)$, $\chi_{315}(194,·)$, $\chi_{315}(164,·)$, $\chi_{315}(166,·)$, $\chi_{315}(134,·)$, $\chi_{315}(74,·)$, $\chi_{315}(241,·)$, $\chi_{315}(149,·)$, $\chi_{315}(151,·)$, $\chi_{315}(121,·)$, $\chi_{315}(314,·)$, $\chi_{315}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{112} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} + \frac{1}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{224} a^{7} - \frac{1}{16} a^{5} + \frac{7}{32} a^{3} + \frac{1}{8} a$, $\frac{1}{896} a^{8} + \frac{1}{448} a^{6} + \frac{7}{128} a^{4} - \frac{1}{4} a^{3} + \frac{5}{32} a^{2} - \frac{1}{4} a$, $\frac{1}{3584} a^{9} + \frac{1}{3584} a^{8} - \frac{3}{1792} a^{7} + \frac{1}{1792} a^{6} + \frac{7}{512} a^{5} - \frac{9}{512} a^{4} + \frac{11}{128} a^{3} - \frac{7}{128} a^{2} - \frac{5}{16} a$, $\frac{1}{14336} a^{10} + \frac{1}{14336} a^{8} + \frac{1}{1792} a^{7} - \frac{1}{14336} a^{6} + \frac{7}{128} a^{5} + \frac{109}{2048} a^{4} - \frac{9}{256} a^{3} + \frac{119}{512} a^{2} - \frac{15}{64} a - \frac{1}{2}$, $\frac{1}{244837892096} a^{11} + \frac{5266767}{244837892096} a^{10} + \frac{1869911}{34976841728} a^{9} + \frac{26929655}{244837892096} a^{8} + \frac{168670455}{244837892096} a^{7} + \frac{968187137}{244837892096} a^{6} + \frac{1166041373}{34976841728} a^{5} + \frac{1972444091}{34976841728} a^{4} + \frac{375811353}{8744210432} a^{3} + \frac{2100183969}{8744210432} a^{2} - \frac{336640025}{1093026304} a - \frac{1521033}{34157072}$

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

Class group and class number

$C_{1554}$, which has order $1554$ (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:  \( 527982.4344714497 \) (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

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-15}) \), 3.3.3969.1, \(\Q(\sqrt{-7}, \sqrt{-15})\), 6.0.110270727.2, 6.6.41351522625.1, 6.0.5907360375.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{12}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$3$3.12.18.51$x^{12} + 27 x^{11} + 15 x^{10} + 36 x^{9} - 36 x^{8} - 18 x^{7} + 21 x^{6} - 18 x^{4} + 27 x^{2} - 27 x + 36$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.10.2$x^{12} + 35 x^{6} + 441$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$