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Properties

Label 12.12.1107856128...8125.1
Degree $12$
Signature $[12, 0]$
Discriminant $3^{18}\cdot 5^{9}\cdot 11^{4}$
Root discriminant $38.64$
Ramified primes $3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_3:S_3.C_2$ (as 12T73)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3509, -11253, -4356, 13904, 9900, -4653, -4122, 531, 639, -19, -42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 42*x^10 - 19*x^9 + 639*x^8 + 531*x^7 - 4122*x^6 - 4653*x^5 + 9900*x^4 + 13904*x^3 - 4356*x^2 - 11253*x - 3509)
 
gp: K = bnfinit(x^12 - 42*x^10 - 19*x^9 + 639*x^8 + 531*x^7 - 4122*x^6 - 4653*x^5 + 9900*x^4 + 13904*x^3 - 4356*x^2 - 11253*x - 3509, 1)
 

Normalized defining polynomial

\( x^{12} - 42 x^{10} - 19 x^{9} + 639 x^{8} + 531 x^{7} - 4122 x^{6} - 4653 x^{5} + 9900 x^{4} + 13904 x^{3} - 4356 x^{2} - 11253 x - 3509 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11078561287986328125=3^{18}\cdot 5^{9}\cdot 11^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.64$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{33} a^{9} + \frac{2}{33} a^{7} + \frac{1}{11} a^{6} + \frac{4}{11} a^{5} + \frac{14}{33} a^{4} + \frac{1}{11} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{33} a^{10} + \frac{2}{33} a^{8} + \frac{1}{11} a^{7} + \frac{1}{33} a^{6} + \frac{14}{33} a^{5} + \frac{1}{11} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{472870497} a^{11} - \frac{2952715}{472870497} a^{10} - \frac{2423141}{472870497} a^{9} - \frac{2704330}{42988227} a^{8} - \frac{9007430}{472870497} a^{7} + \frac{5389723}{157623499} a^{6} - \frac{194282939}{472870497} a^{5} + \frac{49918450}{472870497} a^{4} + \frac{32046226}{157623499} a^{3} - \frac{6398629}{14329409} a^{2} - \frac{8146100}{42988227} a + \frac{9313030}{42988227}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 387641.416817 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3.C_2$ (as 12T73):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 18 conjugacy class representatives for $C_3\times C_3:S_3.C_2$
Character table for $C_3\times C_3:S_3.C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 27 sibling: data not computed
Degree 36 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.12.0.1}{12} }$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }$ R ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$3$3.12.18.69$x^{12} - 39 x^{11} + 15 x^{10} - 30 x^{9} - 36 x^{8} - 9 x^{7} + 30 x^{6} - 27 x^{5} - 36 x^{3} + 27 x + 18$$6$$2$$18$12T73$[3/2, 3/2, 2]_{2}^{2}$
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$11$11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$