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Properties

Label 12.0.11686116501...8848.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 3^{16}\cdot 13^{9}$
Root discriminant $47.02$
Ramified primes $2, 3, 13$
Class number $108$
Class group $[3, 6, 6]$
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![145232, 181776, 58320, 29864, 20736, -216, 2484, -36, 216, -2, 24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 24*x^10 - 2*x^9 + 216*x^8 - 36*x^7 + 2484*x^6 - 216*x^5 + 20736*x^4 + 29864*x^3 + 58320*x^2 + 181776*x + 145232)
 
gp: K = bnfinit(x^12 + 24*x^10 - 2*x^9 + 216*x^8 - 36*x^7 + 2484*x^6 - 216*x^5 + 20736*x^4 + 29864*x^3 + 58320*x^2 + 181776*x + 145232, 1)
 

Normalized defining polynomial

\( x^{12} + 24 x^{10} - 2 x^{9} + 216 x^{8} - 36 x^{7} + 2484 x^{6} - 216 x^{5} + 20736 x^{4} + 29864 x^{3} + 58320 x^{2} + 181776 x + 145232 \)

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:  \(116861165018676718848=2^{8}\cdot 3^{16}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.02$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{108} a^{6} + \frac{1}{9} a^{4} + \frac{1}{3} a^{2} - \frac{7}{27}$, $\frac{1}{108} a^{7} + \frac{1}{9} a^{5} - \frac{1}{6} a^{3} - \frac{7}{27} a$, $\frac{1}{108} a^{8} - \frac{7}{27} a^{2} + \frac{1}{9}$, $\frac{1}{169128} a^{9} + \frac{1}{9396} a^{7} + \frac{337}{84564} a^{6} + \frac{1}{1566} a^{5} + \frac{337}{7047} a^{4} + \frac{4330}{21141} a^{3} + \frac{337}{2349} a^{2} + \frac{1559}{7047} a - \frac{167}{729}$, $\frac{1}{169128} a^{10} + \frac{1}{9396} a^{8} + \frac{337}{84564} a^{7} + \frac{1}{1566} a^{6} + \frac{337}{7047} a^{5} + \frac{4330}{21141} a^{4} + \frac{337}{2349} a^{3} + \frac{1559}{7047} a^{2} - \frac{167}{729} a$, $\frac{1}{230183208} a^{11} + \frac{157}{76727736} a^{10} + \frac{11}{115091604} a^{9} - \frac{34091}{28772901} a^{8} - \frac{50939}{38363868} a^{7} - \frac{23222}{28772901} a^{6} + \frac{581572}{28772901} a^{5} + \frac{1418189}{9590967} a^{4} - \frac{10720771}{57545802} a^{3} + \frac{12463541}{28772901} a^{2} + \frac{459274}{9590967} a - \frac{413309}{992169}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{6}$, which has order $108$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3416.07942432 \)
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

\(\Q(\sqrt{13}) \), 3.3.4212.1 x3, 4.0.2197.1, 6.6.230632272.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$$[\ ]_{3}^{4}$
$3$3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$