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Properties

Label 12.0.1803751953125.2
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 31^{4}$
Root discriminant $10.50$
Ramified primes $5, 31$
Class number $1$
Class group Trivial
Galois group $(C_3\times C_3):C_4$ (as 12T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -30, 75, -110, 126, -120, 106, -85, 61, -35, 16, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 5*x^11 + 16*x^10 - 35*x^9 + 61*x^8 - 85*x^7 + 106*x^6 - 120*x^5 + 126*x^4 - 110*x^3 + 75*x^2 - 30*x + 5)
 
gp: K = bnfinit(x^12 - 5*x^11 + 16*x^10 - 35*x^9 + 61*x^8 - 85*x^7 + 106*x^6 - 120*x^5 + 126*x^4 - 110*x^3 + 75*x^2 - 30*x + 5, 1)
 

Normalized defining polynomial

\( x^{12} - 5 x^{11} + 16 x^{10} - 35 x^{9} + 61 x^{8} - 85 x^{7} + 106 x^{6} - 120 x^{5} + 126 x^{4} - 110 x^{3} + 75 x^{2} - 30 x + 5 \)

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:  \(1803751953125=5^{9}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.50$
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:  $5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{60} a^{10} + \frac{2}{15} a^{9} + \frac{3}{20} a^{8} - \frac{1}{10} a^{7} + \frac{1}{15} a^{6} + \frac{13}{60} a^{5} + \frac{11}{60} a^{4} - \frac{1}{3} a^{3} + \frac{1}{12} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{60} a^{11} + \frac{1}{12} a^{9} + \frac{1}{30} a^{8} - \frac{7}{15} a^{7} - \frac{19}{60} a^{6} + \frac{9}{20} a^{5} - \frac{7}{15} a^{4} + \frac{1}{12} a^{3} + \frac{1}{4} a^{2} - \frac{5}{12} a - \frac{1}{3}$

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:  \( \frac{3}{10} a^{11} - \frac{79}{60} a^{10} + \frac{119}{30} a^{9} - \frac{95}{12} a^{8} + \frac{79}{6} a^{7} - \frac{509}{30} a^{6} + \frac{1259}{60} a^{5} - \frac{451}{20} a^{4} + \frac{139}{6} a^{3} - \frac{209}{12} a^{2} + \frac{133}{12} a - \frac{25}{12} \) (order $10$)
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:  \( 75.4332875798 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3.C_2$ (as 12T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 6 conjugacy class representatives for $(C_3\times C_3):C_4$
Character table for $(C_3\times C_3):C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 6.2.600625.1 x3

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 6 siblings: 6.2.600625.1, 6.2.577200625.1
Degree 9 sibling: 9.1.13867245015625.1
Degree 12 sibling: Deg 12
Degree 18 sibling: Deg 18

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
31Data not computed