↑

Properties

Label 12.12.7370704507...3568.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{18}\cdot 7^{8}\cdot 53^{3}\cdot 181^{2}$
Root discriminant $66.42$
Ramified primes $2, 7, 53, 181$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 12T205

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-122123, -313842, -203954, 67246, 100217, 5452, -16998, -2246, 1386, 172, -58, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 58*x^10 + 172*x^9 + 1386*x^8 - 2246*x^7 - 16998*x^6 + 5452*x^5 + 100217*x^4 + 67246*x^3 - 203954*x^2 - 313842*x - 122123)
 
gp: K = bnfinit(x^12 - 4*x^11 - 58*x^10 + 172*x^9 + 1386*x^8 - 2246*x^7 - 16998*x^6 + 5452*x^5 + 100217*x^4 + 67246*x^3 - 203954*x^2 - 313842*x - 122123, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} - 58 x^{10} + 172 x^{9} + 1386 x^{8} - 2246 x^{7} - 16998 x^{6} + 5452 x^{5} + 100217 x^{4} + 67246 x^{3} - 203954 x^{2} - 313842 x - 122123 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7370704507157871853568=2^{18}\cdot 7^{8}\cdot 53^{3}\cdot 181^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.42$
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:  $2, 7, 53, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{30369602131670081} a^{11} + \frac{8048655380880367}{30369602131670081} a^{10} - \frac{1850076707376895}{30369602131670081} a^{9} + \frac{10855562181665689}{30369602131670081} a^{8} - \frac{14958564049259848}{30369602131670081} a^{7} - \frac{589557756865303}{4338514590238583} a^{6} - \frac{12507901676332379}{30369602131670081} a^{5} + \frac{11541286887438955}{30369602131670081} a^{4} - \frac{3004245139544993}{30369602131670081} a^{3} + \frac{4334801849661065}{30369602131670081} a^{2} + \frac{1941030641178761}{4338514590238583} a - \frac{2955569547058559}{30369602131670081}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 6359125.3416 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T205:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 24 conjugacy class representatives for [E(4)^3:3:2]3
Character table for [E(4)^3:3:2]3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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

Sibling fields

Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Degree 36 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 ${\href{/LocalNumberField/3.12.0.1}{12} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.18.58$x^{12} + 4 x^{11} + 6 x^{10} + 4 x^{9} - 4 x^{7} - 4 x^{6} + 8 x^{5} - 4 x^{4} + 8 x^{3} + 8 x^{2} + 8$$4$$3$$18$12T99$[2, 2, 2, 2, 2]^{6}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
181Data not computed