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Properties

Label 12.0.14354608395...5536.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{18}\cdot 67^{6}$
Root discriminant $85.06$
Ramified primes $2, 3, 67$
Class number $10062$ (GRH)
Class group $[10062]$ (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![45581077, -11755602, 13126680, -2834218, 1651035, -291360, 115139, -15990, 4686, -470, 105, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 105*x^10 - 470*x^9 + 4686*x^8 - 15990*x^7 + 115139*x^6 - 291360*x^5 + 1651035*x^4 - 2834218*x^3 + 13126680*x^2 - 11755602*x + 45581077)
 
gp: K = bnfinit(x^12 - 6*x^11 + 105*x^10 - 470*x^9 + 4686*x^8 - 15990*x^7 + 115139*x^6 - 291360*x^5 + 1651035*x^4 - 2834218*x^3 + 13126680*x^2 - 11755602*x + 45581077, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 105 x^{10} - 470 x^{9} + 4686 x^{8} - 15990 x^{7} + 115139 x^{6} - 291360 x^{5} + 1651035 x^{4} - 2834218 x^{3} + 13126680 x^{2} - 11755602 x + 45581077 \)

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:  \(143546083959041477185536=2^{12}\cdot 3^{18}\cdot 67^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.06$
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:  $2, 3, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2412=2^{2}\cdot 3^{2}\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{2412}(1,·)$, $\chi_{2412}(1475,·)$, $\chi_{2412}(133,·)$, $\chi_{2412}(1607,·)$, $\chi_{2412}(1609,·)$, $\chi_{2412}(2279,·)$, $\chi_{2412}(1741,·)$, $\chi_{2412}(803,·)$, $\chi_{2412}(937,·)$, $\chi_{2412}(2411,·)$, $\chi_{2412}(671,·)$, $\chi_{2412}(805,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{127059498757} a^{10} - \frac{5}{127059498757} a^{9} + \frac{33708838981}{127059498757} a^{8} - \frac{7775857137}{127059498757} a^{7} - \frac{15686305127}{127059498757} a^{6} + \frac{10744665961}{127059498757} a^{5} + \frac{9377718091}{127059498757} a^{4} - \frac{24558462974}{127059498757} a^{3} + \frac{1229252804}{127059498757} a^{2} - \frac{7039850595}{127059498757} a + \frac{59235308655}{127059498757}$, $\frac{1}{53153943650504623} a^{11} + \frac{209164}{53153943650504623} a^{10} - \frac{18823449854560143}{53153943650504623} a^{9} + \frac{12025469860293716}{53153943650504623} a^{8} + \frac{6163788039994904}{53153943650504623} a^{7} - \frac{13896390895595824}{53153943650504623} a^{6} + \frac{2453169740602083}{53153943650504623} a^{5} + \frac{13015933867769919}{53153943650504623} a^{4} + \frac{900364170883053}{53153943650504623} a^{3} - \frac{1900228689485822}{53153943650504623} a^{2} + \frac{16217526500145182}{53153943650504623} a + \frac{4312128523583906}{53153943650504623}$

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

Class group and class number

$C_{10062}$, which has order $10062$ (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:  \( 325.67540279491664 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-67}) \), \(\Q(\sqrt{-201}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{3}, \sqrt{-67})\), \(\Q(\zeta_{36})^+\), 6.0.1973306043.4, 6.0.378874760256.5

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.12.18.82$x^{12} - 9 x^{9} + 9 x^{8} - 9 x^{5} - 9 x^{4} - 9 x^{3} + 9$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
$67$67.12.6.1$x^{12} + 8978 x^{8} + 7218312 x^{6} + 20151121 x^{4} + 31052877461 x^{2} + 13026007032336$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$