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Properties

Label 12.0.51984695918...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 5^{6}\cdot 37^{9}$
Root discriminant $53.25$
Ramified primes $2, 5, 37$
Class number $640$ (GRH)
Class group $[2, 2, 2, 4, 20]$ (GRH)
Galois group $C_3 : C_4$ (as 12T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![531441, 610173, 712233, 330183, 212283, 44886, 27041, 646, 2333, -87, 88, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 88*x^10 - 87*x^9 + 2333*x^8 + 646*x^7 + 27041*x^6 + 44886*x^5 + 212283*x^4 + 330183*x^3 + 712233*x^2 + 610173*x + 531441)
 
gp: K = bnfinit(x^12 - 2*x^11 + 88*x^10 - 87*x^9 + 2333*x^8 + 646*x^7 + 27041*x^6 + 44886*x^5 + 212283*x^4 + 330183*x^3 + 712233*x^2 + 610173*x + 531441, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 88 x^{10} - 87 x^{9} + 2333 x^{8} + 646 x^{7} + 27041 x^{6} + 44886 x^{5} + 212283 x^{4} + 330183 x^{3} + 712233 x^{2} + 610173 x + 531441 \)

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:  \(519846959180308000000=2^{8}\cdot 5^{6}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.25$
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:  $2, 5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{2}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{9} a^{2}$, $\frac{1}{27} a^{9} + \frac{1}{27} a^{8} + \frac{1}{27} a^{7} - \frac{4}{9} a^{6} - \frac{7}{27} a^{5} + \frac{4}{27} a^{4} + \frac{8}{27} a^{3} - \frac{1}{3} a$, $\frac{1}{11380419} a^{10} + \frac{142180}{11380419} a^{9} - \frac{447005}{11380419} a^{8} + \frac{429280}{3793473} a^{7} + \frac{4524482}{11380419} a^{6} + \frac{3872176}{11380419} a^{5} + \frac{5094671}{11380419} a^{4} - \frac{202289}{3793473} a^{3} - \frac{508475}{1264491} a^{2} + \frac{138499}{421497} a + \frac{7000}{15611}$, $\frac{1}{2898366652788522997203} a^{11} - \frac{63923260313441}{2898366652788522997203} a^{10} - \frac{48565789191667494506}{2898366652788522997203} a^{9} - \frac{45353211507418359899}{966122217596174332401} a^{8} - \frac{15252998293454366527}{2898366652788522997203} a^{7} - \frac{792239249500633564616}{2898366652788522997203} a^{6} - \frac{377243249060186314906}{2898366652788522997203} a^{5} + \frac{225252853646441396845}{966122217596174332401} a^{4} - \frac{56352515924865943253}{322040739198724777467} a^{3} + \frac{50747171032794648922}{107346913066241592489} a^{2} + \frac{1547474087079388300}{3975811595045984907} a + \frac{538405226800366}{2197795243253723}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{20}$, which has order $640$ (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:  \( 220.342128705 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:C_4$ (as 12T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $C_3 : C_4$
Character table for $C_3 : C_4$

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.148.1 x3, 4.0.1266325.1, 6.6.810448.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$37$37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$