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Properties

Label 12.0.15088690771...0592.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{16}\cdot 11^{6}\cdot 37^{9}$
Root discriminant $125.38$
Ramified primes $2, 11, 37$
Class number $131648$ (GRH)
Class group $[2, 2, 2, 22, 748]$ (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![6688402169, -2287314156, 2512428095, -392841486, 177880768, -14897420, 4913521, -232812, 64702, -1618, 409, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 409*x^10 - 1618*x^9 + 64702*x^8 - 232812*x^7 + 4913521*x^6 - 14897420*x^5 + 177880768*x^4 - 392841486*x^3 + 2512428095*x^2 - 2287314156*x + 6688402169)
 
gp: K = bnfinit(x^12 - 4*x^11 + 409*x^10 - 1618*x^9 + 64702*x^8 - 232812*x^7 + 4913521*x^6 - 14897420*x^5 + 177880768*x^4 - 392841486*x^3 + 2512428095*x^2 - 2287314156*x + 6688402169, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 409 x^{10} - 1618 x^{9} + 64702 x^{8} - 232812 x^{7} + 4913521 x^{6} - 14897420 x^{5} + 177880768 x^{4} - 392841486 x^{3} + 2512428095 x^{2} - 2287314156 x + 6688402169 \)

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:  \(15088690771598141370990592=2^{16}\cdot 11^{6}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $125.38$
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:  $2, 11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11} a^{6} - \frac{2}{11} a^{5} - \frac{1}{11} a^{4} + \frac{3}{11} a^{3} - \frac{1}{11} a + \frac{3}{11}$, $\frac{1}{11} a^{7} - \frac{5}{11} a^{5} + \frac{1}{11} a^{4} - \frac{5}{11} a^{3} - \frac{1}{11} a^{2} + \frac{1}{11} a - \frac{5}{11}$, $\frac{1}{407} a^{8} + \frac{7}{407} a^{7} - \frac{8}{407} a^{6} - \frac{28}{407} a^{5} + \frac{16}{407} a^{4} + \frac{32}{407} a^{3} + \frac{93}{407} a^{2} + \frac{93}{407} a - \frac{9}{37}$, $\frac{1}{1221} a^{9} + \frac{18}{407} a^{7} + \frac{28}{1221} a^{6} - \frac{343}{1221} a^{5} + \frac{31}{1221} a^{4} + \frac{128}{1221} a^{3} + \frac{184}{407} a^{2} + \frac{175}{1221} a + \frac{545}{1221}$, $\frac{1}{1221} a^{10} - \frac{17}{1221} a^{7} - \frac{2}{111} a^{6} + \frac{100}{1221} a^{5} - \frac{292}{1221} a^{4} + \frac{163}{407} a^{3} - \frac{8}{33} a^{2} - \frac{10}{33} a - \frac{105}{407}$, $\frac{1}{2195197449895964586635733879994388631} a^{11} + \frac{764629981400805583603714154937254}{2195197449895964586635733879994388631} a^{10} + \frac{82453054357813567747253734390607}{731732483298654862211911293331462877} a^{9} - \frac{2369520835002274433654342903037599}{2195197449895964586635733879994388631} a^{8} - \frac{16916568239188089661645252763175170}{2195197449895964586635733879994388631} a^{7} + \frac{86165803772360833643030717392849487}{2195197449895964586635733879994388631} a^{6} - \frac{51405471785912167397435172010116148}{199563404535996780603248534544944421} a^{5} + \frac{1049610113748515828910757399569515422}{2195197449895964586635733879994388631} a^{4} + \frac{794258043107251628004716355003580969}{2195197449895964586635733879994388631} a^{3} - \frac{60137951913416260877437637465305370}{2195197449895964586635733879994388631} a^{2} - \frac{896187434094641303087377187009264081}{2195197449895964586635733879994388631} a - \frac{194646656911569584287379198403386619}{731732483298654862211911293331462877}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{22}\times C_{748}$, which has order $131648$ (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, 4.0.98064208.2, 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.12.16.18$x^{12} + x^{10} + 6 x^{8} - 3 x^{6} + 6 x^{4} + x^{2} - 3$$6$$2$$16$$C_3 : C_4$$[2]_{3}^{2}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$