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Properties

Label 12.0.42516909149...4144.4
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 7^{6}\cdot 13^{10}$
Root discriminant $63.44$
Ramified primes $2, 7, 13$
Class number $1176$ (GRH)
Class group $[7, 168]$ (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![341056, 0, 80832, 0, 16464, 0, 136, 0, 304, 0, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 18*x^10 + 304*x^8 + 136*x^6 + 16464*x^4 + 80832*x^2 + 341056)
 
gp: K = bnfinit(x^12 - 18*x^10 + 304*x^8 + 136*x^6 + 16464*x^4 + 80832*x^2 + 341056, 1)
 

Normalized defining polynomial

\( x^{12} - 18 x^{10} + 304 x^{8} + 136 x^{6} + 16464 x^{4} + 80832 x^{2} + 341056 \)

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:  \(4251690914950152454144=2^{18}\cdot 7^{6}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.44$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(728=2^{3}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{728}(1,·)$, $\chi_{728}(701,·)$, $\chi_{728}(545,·)$, $\chi_{728}(393,·)$, $\chi_{728}(685,·)$, $\chi_{728}(237,·)$, $\chi_{728}(113,·)$, $\chi_{728}(309,·)$, $\chi_{728}(433,·)$, $\chi_{728}(153,·)$, $\chi_{728}(573,·)$, $\chi_{728}(589,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{20} a^{4} + \frac{1}{5}$, $\frac{1}{20} a^{5} + \frac{1}{5} a$, $\frac{1}{40} a^{6} + \frac{1}{10} a^{2}$, $\frac{1}{40} a^{7} + \frac{1}{10} a^{3}$, $\frac{1}{400} a^{8} + \frac{1}{50} a^{4} + \frac{1}{25}$, $\frac{1}{400} a^{9} + \frac{1}{50} a^{5} + \frac{1}{25} a$, $\frac{1}{23500000} a^{10} + \frac{1959}{5875000} a^{8} - \frac{35813}{5875000} a^{6} + \frac{17154}{734375} a^{4} - \frac{206439}{1468750} a^{2} - \frac{227177}{734375}$, $\frac{1}{1715500000} a^{11} - \frac{3427}{53609375} a^{9} + \frac{982453}{107218750} a^{7} + \frac{1272991}{214437500} a^{5} - \frac{21943939}{107218750} a^{3} - \frac{26135927}{53609375} a$

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

Class group and class number

$C_{7}\times C_{168}$, which has order $1176$ (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:  \( 4543.270357084286 \) (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

\(\Q(\sqrt{26}) \), \(\Q(\sqrt{-91}) \), \(\Q(\sqrt{-14}) \), 3.3.169.1, \(\Q(\sqrt{-14}, \sqrt{26})\), 6.6.190102016.1, 6.0.127353499.1, 6.0.5015768576.6

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$7$7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$