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Properties

Label 12.12.1002380611...0000.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{16}\cdot 5^{9}\cdot 23^{8}$
Root discriminant $68.14$
Ramified primes $2, 5, 23$
Class number $1$ (GRH)
Class group Trivial (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![3581, 123410, 55283, -166988, 1674, 56982, -8589, -6896, 1442, 294, -69, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 69*x^10 + 294*x^9 + 1442*x^8 - 6896*x^7 - 8589*x^6 + 56982*x^5 + 1674*x^4 - 166988*x^3 + 55283*x^2 + 123410*x + 3581)
 
gp: K = bnfinit(x^12 - 4*x^11 - 69*x^10 + 294*x^9 + 1442*x^8 - 6896*x^7 - 8589*x^6 + 56982*x^5 + 1674*x^4 - 166988*x^3 + 55283*x^2 + 123410*x + 3581, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} - 69 x^{10} + 294 x^{9} + 1442 x^{8} - 6896 x^{7} - 8589 x^{6} + 56982 x^{5} + 1674 x^{4} - 166988 x^{3} + 55283 x^{2} + 123410 x + 3581 \)

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:  \(10023806115968000000000=2^{16}\cdot 5^{9}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.14$
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:  $2, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{10} a^{4} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{10}$, $\frac{1}{10} a^{9} - \frac{1}{5} a^{7} + \frac{1}{10} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{3}{10} a^{3} + \frac{2}{5} a + \frac{3}{10}$, $\frac{1}{39064790} a^{10} - \frac{976483}{39064790} a^{9} + \frac{395163}{39064790} a^{8} + \frac{8701957}{39064790} a^{7} - \frac{1759566}{19532395} a^{6} + \frac{3278891}{7812958} a^{5} - \frac{5869933}{19532395} a^{4} - \frac{1107892}{19532395} a^{3} - \frac{5349313}{19532395} a^{2} - \frac{3296752}{19532395} a - \frac{1482989}{39064790}$, $\frac{1}{14172510488050} a^{11} - \frac{1828}{283450209761} a^{10} - \frac{258260580152}{7086255244025} a^{9} + \frac{96512502524}{7086255244025} a^{8} + \frac{1209384945757}{7086255244025} a^{7} + \frac{472680535781}{2834502097610} a^{6} - \frac{3158762683007}{7086255244025} a^{5} - \frac{5923476628859}{14172510488050} a^{4} + \frac{417761161453}{14172510488050} a^{3} - \frac{3672944279691}{14172510488050} a^{2} + \frac{3254069006909}{14172510488050} a - \frac{229370478747}{7086255244025}$

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:  \( 15635866.4176 \) (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

\(\Q(\sqrt{5}) \), 3.3.10580.1 x3, \(\Q(\zeta_{20})^+\), 6.6.559682000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

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$$[2]_{3}^{2}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$23$23.12.8.1$x^{12} - 69 x^{9} + 1587 x^{6} - 12167 x^{3} + 372468371$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$