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Properties

Label 12.0.73358639104.1
Degree $12$
Signature $[0, 6]$
Discriminant $73358639104$
Root discriminant $8.04$
Ramified primes $2, 23$
Class number $1$
Class group trivial
Galois group $C_2 \times S_4$ (as 12T23)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 2*x^10 - 2*x^9 + 6*x^8 - 10*x^7 + 10*x^6 - 12*x^5 + 17*x^4 - 14*x^3 + 8*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^12 - 2*x^11 + 2*x^10 - 2*x^9 + 6*x^8 - 10*x^7 + 10*x^6 - 12*x^5 + 17*x^4 - 14*x^3 + 8*x^2 - 4*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 8, -14, 17, -12, 10, -10, 6, -2, 2, -2, 1]);
 

\( x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{9} + 6 x^{8} - 10 x^{7} + 10 x^{6} - 12 x^{5} + 17 x^{4} - 14 x^{3} + 8 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(73358639104\)\(\medspace = 2^{18}\cdot 23^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $8.04$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{79} a^{11} - \frac{14}{79} a^{10} + \frac{12}{79} a^{9} + \frac{12}{79} a^{8} + \frac{20}{79} a^{7} - \frac{13}{79} a^{6} + \frac{8}{79} a^{5} - \frac{29}{79} a^{4} - \frac{30}{79} a^{3} + \frac{30}{79} a^{2} - \frac{36}{79} a + \frac{33}{79}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{3985}{79} a^{11} - \frac{5546}{79} a^{10} + \frac{4607}{79} a^{9} - \frac{5189}{79} a^{8} + \frac{20766}{79} a^{7} - \frac{27236}{79} a^{6} + \frac{23348}{79} a^{5} - \frac{33721}{79} a^{4} + \frac{47298}{79} a^{3} - \frac{27153}{79} a^{2} + \frac{15567}{79} a - \frac{6587}{79} \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 3.42242551816 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 3.42242551816 \cdot 1}{4\sqrt{73358639104}}\approx 0.194369264472$

Galois group

$C_2\times S_4$ (as 12T23):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 48
The 10 conjugacy class representatives for $C_2 \times S_4$
Character table for $C_2 \times S_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.23.1, 6.0.33856.1, 6.0.33856.2, 6.2.33856.1

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

Sibling fields

Degree 6 siblings: 6.2.778688.4, 6.0.33856.1
Degree 8 siblings: 8.0.8667136.1, 8.4.4584914944.1
Degree 12 siblings: 12.2.1687248699392.2, 12.0.606355001344.2, 12.4.38806720086016.2, 12.0.38806720086016.1, 12.0.38806720086016.3
Degree 16 sibling: Deg 16
Degree 24 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.12.18.51$x^{12} + 10 x^{11} + 16 x^{10} + 16 x^{9} - 6 x^{8} + 16 x^{7} - 8 x^{6} - 8 x^{5} + 4 x^{4} - 8 x^{3} + 16 x + 8$$4$$3$$18$$A_4 \times C_2$$[2, 2, 2]^{3}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$