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Properties

Label 12.0.30758641052...2697.1
Degree $12$
Signature $[0, 6]$
Discriminant $11^{10}\cdot 17^{9}$
Root discriminant $61.75$
Ramified primes $11, 17$
Class number $136$ (GRH)
Class group $[2, 2, 34]$ (GRH)
Galois group $C_3 : C_4$ (as 12T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![332929, 51353, 86932, 34538, 17286, -7280, 2315, -316, 278, -81, 41, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 5*x^11 + 41*x^10 - 81*x^9 + 278*x^8 - 316*x^7 + 2315*x^6 - 7280*x^5 + 17286*x^4 + 34538*x^3 + 86932*x^2 + 51353*x + 332929)
 
gp: K = bnfinit(x^12 - 5*x^11 + 41*x^10 - 81*x^9 + 278*x^8 - 316*x^7 + 2315*x^6 - 7280*x^5 + 17286*x^4 + 34538*x^3 + 86932*x^2 + 51353*x + 332929, 1)
 

Normalized defining polynomial

\( x^{12} - 5 x^{11} + 41 x^{10} - 81 x^{9} + 278 x^{8} - 316 x^{7} + 2315 x^{6} - 7280 x^{5} + 17286 x^{4} + 34538 x^{3} + 86932 x^{2} + 51353 x + 332929 \)

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:  \(3075864105233637502697=11^{10}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.75$
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:  $11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{12} a^{6} + \frac{1}{12} a^{5} - \frac{1}{12} a^{4} - \frac{1}{12} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{4}$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{1}{4} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{12} a^{3} + \frac{1}{3} a^{2} + \frac{1}{12} a - \frac{5}{12}$, $\frac{1}{12998453243374605722925756} a^{11} - \frac{89728906460959003748816}{3249613310843651430731439} a^{10} - \frac{113960419188700578926353}{6499226621687302861462878} a^{9} - \frac{355256658616009211543605}{6499226621687302861462878} a^{8} + \frac{634976056959576799014632}{3249613310843651430731439} a^{7} - \frac{311603942657777302620263}{4332817747791535240975252} a^{6} + \frac{3559144666817302568877521}{12998453243374605722925756} a^{5} + \frac{1074968851944094715331597}{2166408873895767620487626} a^{4} + \frac{4347680227433098227104993}{12998453243374605722925756} a^{3} + \frac{1302778922647282054329805}{4332817747791535240975252} a^{2} - \frac{3394759806540702505726379}{12998453243374605722925756} a - \frac{1359629699319349914975}{7509216200678570608276}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{34}$, which has order $136$ (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:  \( 3083.51591562 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:C_4$ (as 12T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $C_3 : C_4$
Character table for $C_3 : C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 3.3.2057.1 x3, 4.0.594473.1, 6.6.71931233.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$11$11.12.10.3$x^{12} + 220 x^{6} + 41503$$6$$2$$10$$C_3 : C_4$$[\ ]_{6}^{2}$
$17$17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$