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Properties

Label 12.0.64015942333...2928.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 17^{9}$
Root discriminant $207.58$
Ramified primes $2, 3, 7, 17$
Class number $1077280$ (GRH)
Class group $[2, 2, 2, 134660]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![718836830208, 0, 59903069184, 0, 1835265600, 0, 26655048, 0, 197064, 0, 714, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 714*x^10 + 197064*x^8 + 26655048*x^6 + 1835265600*x^4 + 59903069184*x^2 + 718836830208)
 
gp: K = bnfinit(x^12 + 714*x^10 + 197064*x^8 + 26655048*x^6 + 1835265600*x^4 + 59903069184*x^2 + 718836830208, 1)
 

Normalized defining polynomial

\( x^{12} + 714 x^{10} + 197064 x^{8} + 26655048 x^{6} + 1835265600 x^{4} + 59903069184 x^{2} + 718836830208 \)

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:  \(6401594233356076787154812928=2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $207.58$
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:  $2, 3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2856=2^{3}\cdot 3\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{2856}(1,·)$, $\chi_{2856}(2435,·)$, $\chi_{2856}(1633,·)$, $\chi_{2856}(1801,·)$, $\chi_{2856}(395,·)$, $\chi_{2856}(2209,·)$, $\chi_{2856}(1067,·)$, $\chi_{2856}(1475,·)$, $\chi_{2856}(169,·)$, $\chi_{2856}(2041,·)$, $\chi_{2856}(251,·)$, $\chi_{2856}(803,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{6} a^{2}$, $\frac{1}{6} a^{3}$, $\frac{1}{612} a^{4}$, $\frac{1}{612} a^{5}$, $\frac{1}{25704} a^{6}$, $\frac{1}{51408} a^{7} - \frac{1}{1224} a^{5} - \frac{1}{2} a$, $\frac{1}{10487232} a^{8} - \frac{1}{102816} a^{6} - \frac{1}{1224} a^{4} + \frac{1}{24} a^{2}$, $\frac{1}{20974464} a^{9} - \frac{1}{205632} a^{7} - \frac{1}{2448} a^{5} - \frac{1}{16} a^{3} - \frac{1}{2} a$, $\frac{1}{3272016384} a^{10} + \frac{1}{77905152} a^{8} - \frac{5}{891072} a^{6} + \frac{61}{127296} a^{4} - \frac{5}{312} a^{2} + \frac{1}{13}$, $\frac{1}{6544032768} a^{11} + \frac{1}{155810304} a^{9} - \frac{5}{1782144} a^{7} + \frac{61}{254592} a^{5} - \frac{5}{624} a^{3} + \frac{1}{26} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{134660}$, which has order $1077280$ (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:  \( 1059.5454270327698 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.0.138664512.3, 6.6.11796113.1

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 ${\href{/LocalNumberField/5.12.0.1}{12} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$7$7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$17$17.12.9.1$x^{12} - 34 x^{8} - 10115 x^{4} - 397953$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$