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Properties

Label 12.4.82247526720703125.1
Degree $12$
Signature $[4, 4]$
Discriminant $3^{4}\cdot 5^{9}\cdot 151^{4}$
Root discriminant $25.68$
Ramified primes $3, 5, 151$
Class number $1$
Class group Trivial
Galois group 12T159

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, -117, -396, -201, 38, 34, -60, -29, 21, -2, 6, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 5*x^11 + 6*x^10 - 2*x^9 + 21*x^8 - 29*x^7 - 60*x^6 + 34*x^5 + 38*x^4 - 201*x^3 - 396*x^2 - 117*x - 9)
 
gp: K = bnfinit(x^12 - 5*x^11 + 6*x^10 - 2*x^9 + 21*x^8 - 29*x^7 - 60*x^6 + 34*x^5 + 38*x^4 - 201*x^3 - 396*x^2 - 117*x - 9, 1)
 

Normalized defining polynomial

\( x^{12} - 5 x^{11} + 6 x^{10} - 2 x^{9} + 21 x^{8} - 29 x^{7} - 60 x^{6} + 34 x^{5} + 38 x^{4} - 201 x^{3} - 396 x^{2} - 117 x - 9 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(82247526720703125=3^{4}\cdot 5^{9}\cdot 151^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.68$
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:  $3, 5, 151$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{1266689341977} a^{11} - \frac{26332157089}{1266689341977} a^{10} + \frac{510294320146}{1266689341977} a^{9} + \frac{253122915283}{1266689341977} a^{8} - \frac{416156034233}{1266689341977} a^{7} - \frac{12385487906}{1266689341977} a^{6} + \frac{452890986316}{1266689341977} a^{5} + \frac{458983346929}{1266689341977} a^{4} + \frac{34205945908}{422229780659} a^{3} + \frac{605836036454}{1266689341977} a^{2} + \frac{133543747372}{422229780659} a - \frac{209044958427}{422229780659}$

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:  $7$
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:  \( 6988.45379584 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T159:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 28 conjugacy class representatives for [2^5]F_18(6)_2
Character table for [2^5]F_18(6)_2 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 6.6.2850125.1

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

Sibling fields

Degree 16 sibling: data not computed
Degree 24 siblings: 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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$151$$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.3.2.1$x^{3} - 151$$3$$1$$2$$C_3$$[\ ]_{3}$
151.3.2.1$x^{3} - 151$$3$$1$$2$$C_3$$[\ ]_{3}$