Properties

Label 12.0.1315127813325481.1
Degree $12$
Signature $[0, 6]$
Discriminant $331^{6}$
Root discriminant $18.19$
Ramified prime $331$
Class number $4$
Class group $[4]$
Galois Group $S_4$ (as 12T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, -175, 280, -269, 315, -315, 288, -198, 108, -50, 21, -6, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 21*x^10 - 50*x^9 + 108*x^8 - 198*x^7 + 288*x^6 - 315*x^5 + 315*x^4 - 269*x^3 + 280*x^2 - 175*x + 125)
gp: K = bnfinit(x^12 - 6*x^11 + 21*x^10 - 50*x^9 + 108*x^8 - 198*x^7 + 288*x^6 - 315*x^5 + 315*x^4 - 269*x^3 + 280*x^2 - 175*x + 125, 1)

Normalized defining polynomial

\(x^{12} \) \(\mathstrut -\mathstrut 6 x^{11} \) \(\mathstrut +\mathstrut 21 x^{10} \) \(\mathstrut -\mathstrut 50 x^{9} \) \(\mathstrut +\mathstrut 108 x^{8} \) \(\mathstrut -\mathstrut 198 x^{7} \) \(\mathstrut +\mathstrut 288 x^{6} \) \(\mathstrut -\mathstrut 315 x^{5} \) \(\mathstrut +\mathstrut 315 x^{4} \) \(\mathstrut -\mathstrut 269 x^{3} \) \(\mathstrut +\mathstrut 280 x^{2} \) \(\mathstrut -\mathstrut 175 x \) \(\mathstrut +\mathstrut 125 \)

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:  \(1315127813325481=331^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $18.19$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $331$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5050} a^{10} - \frac{1}{1010} a^{9} - \frac{199}{5050} a^{8} + \frac{413}{2525} a^{7} - \frac{1731}{5050} a^{6} + \frac{2281}{5050} a^{5} + \frac{517}{2525} a^{4} + \frac{77}{2525} a^{3} + \frac{359}{5050} a^{2} + \frac{233}{505} a + \frac{47}{202}$, $\frac{1}{3014850} a^{11} + \frac{293}{3014850} a^{10} + \frac{9421}{3014850} a^{9} + \frac{32169}{502475} a^{8} + \frac{105039}{1004950} a^{7} - \frac{376889}{1004950} a^{6} + \frac{185172}{502475} a^{5} + \frac{209446}{502475} a^{4} + \frac{411337}{1004950} a^{3} - \frac{318034}{1507425} a^{2} + \frac{41183}{200990} a - \frac{22691}{60297}$

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

Class group and class number

Multiplicative Abelian group isomorphic to C4, order $4$

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:  \( \frac{13391}{1507425} a^{11} - \frac{99023}{1507425} a^{10} + \frac{350801}{1507425} a^{9} - \frac{2749}{4975} a^{8} + \frac{555087}{502475} a^{7} - \frac{1067047}{502475} a^{6} + \frac{1515277}{502475} a^{5} - \frac{1476356}{502475} a^{4} + \frac{1212739}{502475} a^{3} - \frac{4549867}{1507425} a^{2} + \frac{262498}{100495} a - \frac{127940}{60297} \),  \( \frac{33202}{1507425} a^{11} - \frac{314477}{3014850} a^{10} + \frac{882329}{3014850} a^{9} - \frac{539291}{1004950} a^{8} + \frac{549069}{502475} a^{7} - \frac{1682803}{1004950} a^{6} + \frac{1406023}{1004950} a^{5} - \frac{148162}{502475} a^{4} + \frac{290018}{502475} a^{3} + \frac{1017077}{3014850} a^{2} + \frac{70756}{100495} a + \frac{44599}{120594} \),  \( \frac{15531}{502475} a^{11} - \frac{170841}{1004950} a^{10} + \frac{527977}{1004950} a^{9} - \frac{1094589}{1004950} a^{8} + \frac{1104156}{502475} a^{7} - \frac{3816897}{1004950} a^{6} + \frac{4436577}{1004950} a^{5} - \frac{1624518}{502475} a^{4} + \frac{1502757}{502475} a^{3} - \frac{2694219}{1004950} a^{2} + \frac{333254}{100495} a - \frac{30229}{40198} \),  \( \frac{44927}{3014850} a^{11} - \frac{181906}{1507425} a^{10} + \frac{641371}{1507425} a^{9} - \frac{206021}{200990} a^{8} + \frac{2048067}{1004950} a^{7} - \frac{2015971}{502475} a^{6} + \frac{5629077}{1004950} a^{5} - \frac{545027}{100495} a^{4} + \frac{989669}{200990} a^{3} - \frac{16374493}{3014850} a^{2} + \frac{648613}{200990} a - \frac{494045}{120594} \),  \( \frac{833}{3014850} a^{11} - \frac{27514}{1507425} a^{10} + \frac{149314}{1507425} a^{9} - \frac{13187}{40198} a^{8} + \frac{724333}{1004950} a^{7} - \frac{740264}{502475} a^{6} + \frac{2496043}{1004950} a^{5} - \frac{317834}{100495} a^{4} + \frac{498047}{200990} a^{3} - \frac{3498997}{3014850} a^{2} + \frac{25699}{200990} a - \frac{5345}{120594} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 119.503369717 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_4$ (as 12T9):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 24
The 5 conjugacy class representatives for $S_4$
Character table for $S_4$

Intermediate fields

\(\Q(\sqrt{-331}) \), 3.1.331.1 x3, 6.0.36264691.2, 6.2.109561.1, 6.0.36264691.1

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 4 sibling: data not computed
Degree 6 siblings: data not computed
Degree 8 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
331Data not computed