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Properties

Label 12.0.14358355998...000.20
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 5^{6}\cdot 37^{10}$
Root discriminant $222.04$
Ramified primes $2, 3, 5, 37$
Class number $1048416$ (GRH)
Class group $[2, 2, 262104]$ (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![1771822921, -193232084, 248996314, -18766796, 15029687, -732598, 515448, -15346, 11149, -214, 151, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 151*x^10 - 214*x^9 + 11149*x^8 - 15346*x^7 + 515448*x^6 - 732598*x^5 + 15029687*x^4 - 18766796*x^3 + 248996314*x^2 - 193232084*x + 1771822921)
 
gp: K = bnfinit(x^12 - 2*x^11 + 151*x^10 - 214*x^9 + 11149*x^8 - 15346*x^7 + 515448*x^6 - 732598*x^5 + 15029687*x^4 - 18766796*x^3 + 248996314*x^2 - 193232084*x + 1771822921, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 151 x^{10} - 214 x^{9} + 11149 x^{8} - 15346 x^{7} + 515448 x^{6} - 732598 x^{5} + 15029687 x^{4} - 18766796 x^{3} + 248996314 x^{2} - 193232084 x + 1771822921 \)

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:  \(14358355998689738428416000000=2^{18}\cdot 3^{6}\cdot 5^{6}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $222.04$
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:  $2, 3, 5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4440=2^{3}\cdot 3\cdot 5\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{4440}(1,·)$, $\chi_{4440}(4229,·)$, $\chi_{4440}(961,·)$, $\chi_{4440}(841,·)$, $\chi_{4440}(269,·)$, $\chi_{4440}(989,·)$, $\chi_{4440}(4081,·)$, $\chi_{4440}(149,·)$, $\chi_{4440}(1321,·)$, $\chi_{4440}(121,·)$, $\chi_{4440}(1469,·)$, $\chi_{4440}(1109,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11} a^{9} - \frac{2}{11} a^{7} + \frac{4}{11} a^{5} + \frac{3}{11} a^{3} + \frac{5}{11} a$, $\frac{1}{11} a^{10} - \frac{2}{11} a^{8} + \frac{4}{11} a^{6} + \frac{3}{11} a^{4} + \frac{5}{11} a^{2}$, $\frac{1}{62068174914784843796028280113277} a^{11} - \frac{248975698883145803844173137565}{62068174914784843796028280113277} a^{10} + \frac{857934726136571549646863356765}{62068174914784843796028280113277} a^{9} + \frac{2069437309450450318758820418387}{62068174914784843796028280113277} a^{8} + \frac{1402942952232722969116701370382}{62068174914784843796028280113277} a^{7} + \frac{27879963152380845187663967093938}{62068174914784843796028280113277} a^{6} - \frac{8886557157972860789911908274304}{62068174914784843796028280113277} a^{5} + \frac{14471838359585735048105175201762}{62068174914784843796028280113277} a^{4} - \frac{4378218048069219446754978907769}{62068174914784843796028280113277} a^{3} - \frac{3092305388830379483483599657056}{62068174914784843796028280113277} a^{2} + \frac{12861974876716566706873715921328}{62068174914784843796028280113277} a - \frac{4709301819632900622006491076}{37869539301272021840163685243}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{262104}$, which has order $1048416$ (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:  \( 2518.2332404942745 \) (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

\(\Q(\sqrt{-1110}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{37}) \), 3.3.1369.1, \(\Q(\sqrt{-30}, \sqrt{37})\), 6.0.119826357696000.1, 6.0.3238550208000.4, 6.6.69343957.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.18.23$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$$[3]^{6}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$