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Properties

Label 12.12.3914206393...7088.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{8}\cdot 7^{6}\cdot 37^{9}$
Root discriminant $63.01$
Ramified primes $2, 7, 37$
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![531441, -846369, -690363, 785727, 523305, -114510, -100609, 646, 5885, 135, -134, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 134*x^10 + 135*x^9 + 5885*x^8 + 646*x^7 - 100609*x^6 - 114510*x^5 + 523305*x^4 + 785727*x^3 - 690363*x^2 - 846369*x + 531441)
 
gp: K = bnfinit(x^12 - 2*x^11 - 134*x^10 + 135*x^9 + 5885*x^8 + 646*x^7 - 100609*x^6 - 114510*x^5 + 523305*x^4 + 785727*x^3 - 690363*x^2 - 846369*x + 531441, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} - 134 x^{10} + 135 x^{9} + 5885 x^{8} + 646 x^{7} - 100609 x^{6} - 114510 x^{5} + 523305 x^{4} + 785727 x^{3} - 690363 x^{2} - 846369 x + 531441 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3914206393638659577088=2^{8}\cdot 7^{6}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.01$
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:  $2, 7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{4}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{9} a^{4} + \frac{4}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{9} + \frac{1}{27} a^{8} + \frac{4}{27} a^{7} + \frac{4}{9} a^{6} + \frac{8}{27} a^{5} - \frac{5}{27} a^{4} + \frac{5}{27} a^{3} + \frac{4}{9} a^{2}$, $\frac{1}{8305497} a^{10} + \frac{51892}{8305497} a^{9} + \frac{54514}{8305497} a^{8} - \frac{31355}{307611} a^{7} + \frac{3027833}{8305497} a^{6} + \frac{200257}{8305497} a^{5} - \frac{2595031}{8305497} a^{4} - \frac{1366642}{2768499} a^{3} - \frac{299218}{922833} a^{2} + \frac{102685}{307611} a - \frac{3067}{11393}$, $\frac{1}{53227281921289148511} a^{11} + \frac{694356958933}{53227281921289148511} a^{10} - \frac{523968875305922996}{53227281921289148511} a^{9} - \frac{86090376090777205}{1971380811899598093} a^{8} + \frac{6176490166868888753}{53227281921289148511} a^{7} - \frac{4096735206374651987}{53227281921289148511} a^{6} - \frac{15674783657572975003}{53227281921289148511} a^{5} - \frac{2529483857598714940}{17742427307096382837} a^{4} + \frac{1438389283229804165}{5914142435698794279} a^{3} - \frac{402405947589456851}{1971380811899598093} a^{2} + \frac{30201915992058610}{73014104144429559} a + \frac{779647791189831}{2704226079423317}$

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:  \( 4909459.41704 \) (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, 4.4.2481997.1, 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$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}$