↑

Properties

Label 12.0.115273770333632.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{6}\cdot 23^{9}$
Root discriminant $14.85$
Ramified primes $2, 23$
Class number $1$
Class group Trivial
Galois group $(C_6\times C_2):C_2$ (as 12T15)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![216, -72, 408, -68, 306, -84, 201, -99, 97, -38, 19, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 19*x^10 - 38*x^9 + 97*x^8 - 99*x^7 + 201*x^6 - 84*x^5 + 306*x^4 - 68*x^3 + 408*x^2 - 72*x + 216)
 
gp: K = bnfinit(x^12 - 4*x^11 + 19*x^10 - 38*x^9 + 97*x^8 - 99*x^7 + 201*x^6 - 84*x^5 + 306*x^4 - 68*x^3 + 408*x^2 - 72*x + 216, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 19 x^{10} - 38 x^{9} + 97 x^{8} - 99 x^{7} + 201 x^{6} - 84 x^{5} + 306 x^{4} - 68 x^{3} + 408 x^{2} - 72 x + 216 \)

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:  \(115273770333632=2^{6}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.85$
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:  $2, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{14} a^{8} - \frac{2}{7} a^{7} - \frac{3}{14} a^{6} - \frac{2}{7} a^{5} - \frac{5}{14} a^{4} - \frac{1}{2} a^{3} - \frac{1}{14} a^{2} - \frac{3}{7}$, $\frac{1}{14} a^{9} - \frac{5}{14} a^{7} - \frac{1}{7} a^{6} - \frac{1}{2} a^{5} + \frac{1}{14} a^{4} - \frac{1}{14} a^{3} - \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{84} a^{10} + \frac{1}{42} a^{9} + \frac{1}{84} a^{8} - \frac{2}{21} a^{7} - \frac{29}{84} a^{6} - \frac{3}{28} a^{5} - \frac{5}{28} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{21} a$, $\frac{1}{5370372} a^{11} + \frac{12371}{5370372} a^{10} - \frac{156257}{5370372} a^{9} + \frac{37105}{5370372} a^{8} - \frac{184133}{5370372} a^{7} + \frac{81141}{298354} a^{6} - \frac{97634}{447531} a^{5} + \frac{132107}{1790124} a^{4} + \frac{63390}{149177} a^{3} + \frac{131842}{1342593} a^{2} + \frac{162052}{447531} a + \frac{66294}{149177}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( 82.1227317381 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:D_4$ (as 12T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24
The 9 conjugacy class representatives for $(C_6\times C_2):C_2$
Character table for $(C_6\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 4.0.48668.1, 6.0.12167.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$23$23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$