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Properties

Label 12.0.206497759400820736.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{26}\cdot 79^{5}$
Root discriminant $27.73$
Ramified primes $2, 79$
Class number $8$
Class group $[2, 4]$
Galois group $\GL(2,Z/4)$ (as 12T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![227, 918, 1669, 1534, 827, 328, 236, 64, 37, -2, 9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 9*x^10 - 2*x^9 + 37*x^8 + 64*x^7 + 236*x^6 + 328*x^5 + 827*x^4 + 1534*x^3 + 1669*x^2 + 918*x + 227)
 
gp: K = bnfinit(x^12 - 2*x^11 + 9*x^10 - 2*x^9 + 37*x^8 + 64*x^7 + 236*x^6 + 328*x^5 + 827*x^4 + 1534*x^3 + 1669*x^2 + 918*x + 227, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 9 x^{10} - 2 x^{9} + 37 x^{8} + 64 x^{7} + 236 x^{6} + 328 x^{5} + 827 x^{4} + 1534 x^{3} + 1669 x^{2} + 918 x + 227 \)

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:  \(206497759400820736=2^{26}\cdot 79^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.73$
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:  $2, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{10} - \frac{2}{7} a^{8} + \frac{3}{7} a^{7} - \frac{3}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} + \frac{2}{7} a^{2} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{3646007866529} a^{11} + \frac{253236263181}{3646007866529} a^{10} + \frac{26378767620}{520858266647} a^{9} - \frac{98169577978}{520858266647} a^{8} + \frac{1787583736476}{3646007866529} a^{7} + \frac{399711973775}{3646007866529} a^{6} - \frac{11847283356}{520858266647} a^{5} - \frac{566266771880}{3646007866529} a^{4} + \frac{771048248419}{3646007866529} a^{3} - \frac{1761105251548}{3646007866529} a^{2} + \frac{55707983127}{3646007866529} a + \frac{521258229620}{3646007866529}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$

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:  \( 816.348389593 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\GL(2,Z/4)$ (as 12T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 14 conjugacy class representatives for $\GL(2,Z/4)$
Character table for $\GL(2,Z/4)$

Intermediate fields

3.3.316.1, 6.6.3195392.1

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

Sibling fields

Degree 12 siblings: 12.0.2039165374083104768.1, 12.0.825991037603282944.1
Degree 16 siblings: 16.0.4276439742589131330420736.1, Deg 16
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Arithmetically equvalently sibling: 12.0.206497759400820736.2

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.4.6.5$x^{4} + 2 x^{2} - 4$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
2.8.20.85$x^{8} + 8 x^{6} + 100$$8$$1$$20$$C_2^2 \wr C_2$$[2, 2, 3, 7/2]^{2}$
79Data not computed