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Properties

Label 12.0.109436699097976761.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{18}\cdot 7^{10}$
Root discriminant $26.30$
Ramified primes $3, 7$
Class number $21$
Class group $[21]$
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![343, 0, 0, -490, 0, 0, 224, 0, 0, -14, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 14*x^9 + 224*x^6 - 490*x^3 + 343)
 
gp: K = bnfinit(x^12 - 14*x^9 + 224*x^6 - 490*x^3 + 343, 1)
 

Normalized defining polynomial

\( x^{12} - 14 x^{9} + 224 x^{6} - 490 x^{3} + 343 \)

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:  \(109436699097976761=3^{18}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.30$
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:  $3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(63=3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{63}(1,·)$, $\chi_{63}(5,·)$, $\chi_{63}(38,·)$, $\chi_{63}(8,·)$, $\chi_{63}(11,·)$, $\chi_{63}(40,·)$, $\chi_{63}(52,·)$, $\chi_{63}(55,·)$, $\chi_{63}(23,·)$, $\chi_{63}(25,·)$, $\chi_{63}(58,·)$, $\chi_{63}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{28} a^{6} + \frac{1}{4}$, $\frac{1}{28} a^{7} + \frac{1}{4} a$, $\frac{1}{28} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{784} a^{9} + \frac{1}{112} a^{6} - \frac{3}{112} a^{3} - \frac{3}{16}$, $\frac{1}{784} a^{10} + \frac{1}{112} a^{7} - \frac{3}{112} a^{4} - \frac{3}{16} a$, $\frac{1}{5488} a^{11} + \frac{1}{784} a^{8} + \frac{53}{784} a^{5} + \frac{37}{112} a^{2}$

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

Class group and class number

$C_{21}$, which has order $21$

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:  \( \frac{1}{112} a^{9} - \frac{13}{112} a^{6} + \frac{29}{16} a^{3} - \frac{25}{16} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2108.25732612 \)
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

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), 3.3.3969.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), 6.0.110270727.2, 6.6.330812181.2, 6.0.47258883.2

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$$[2]_{2}^{2}$
$7$7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$