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Properties

Label 12.0.62500000000...0000.6
Degree $12$
Signature $[0, 6]$
Discriminant $2^{16}\cdot 5^{20}$
Root discriminant $36.84$
Ramified primes $2, 5$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $S_5$ (as 12T74)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1504, -4224, 7856, -9520, 7390, -3508, 758, 198, -205, 60, -1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - x^10 + 60*x^9 - 205*x^8 + 198*x^7 + 758*x^6 - 3508*x^5 + 7390*x^4 - 9520*x^3 + 7856*x^2 - 4224*x + 1504)
 
gp: K = bnfinit(x^12 - 4*x^11 - x^10 + 60*x^9 - 205*x^8 + 198*x^7 + 758*x^6 - 3508*x^5 + 7390*x^4 - 9520*x^3 + 7856*x^2 - 4224*x + 1504, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} - x^{10} + 60 x^{9} - 205 x^{8} + 198 x^{7} + 758 x^{6} - 3508 x^{5} + 7390 x^{4} - 9520 x^{3} + 7856 x^{2} - 4224 x + 1504 \)

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:  \(6250000000000000000=2^{16}\cdot 5^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.84$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{228} a^{9} + \frac{2}{57} a^{8} - \frac{15}{76} a^{7} - \frac{8}{57} a^{6} + \frac{5}{76} a^{5} - \frac{1}{6} a^{4} + \frac{5}{114} a^{3} - \frac{7}{19} a^{2} - \frac{55}{114} a - \frac{11}{57}$, $\frac{1}{912} a^{10} - \frac{109}{912} a^{8} - \frac{8}{57} a^{7} - \frac{185}{912} a^{6} + \frac{149}{456} a^{5} + \frac{157}{456} a^{4} - \frac{41}{228} a^{3} - \frac{61}{456} a^{2} + \frac{1}{6} a - \frac{13}{114}$, $\frac{1}{164710848} a^{11} - \frac{24541}{82355424} a^{10} + \frac{103817}{54903616} a^{9} - \frac{4591691}{82355424} a^{8} + \frac{28579319}{164710848} a^{7} + \frac{571689}{13725904} a^{6} - \frac{7062691}{27451808} a^{5} - \frac{6617437}{20588856} a^{4} + \frac{5502285}{27451808} a^{3} - \frac{4393283}{13725904} a^{2} - \frac{2799151}{20588856} a - \frac{163213}{541812}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)

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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 46350.8472599 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 12T74):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

\(\Q(\sqrt{-5}) \), 6.0.500000000.1

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

Sibling fields

Degree 5 sibling: data not computed
Degree 6 sibling: data not computed
Degree 10 siblings: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.10.19.2$x^{10} + 80$$10$$1$$19$$F_5$$[9/4]_{4}$