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Properties

Label 12.0.39155492640...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 5^{9}\cdot 23^{8}$
Root discriminant $42.93$
Ramified primes $2, 5, 23$
Class number $16$ (GRH)
Class group $[4, 4]$ (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![83521, -73695, 69938, -61132, 53719, -4632, 3651, -334, 237, -14, 16, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 16*x^10 - 14*x^9 + 237*x^8 - 334*x^7 + 3651*x^6 - 4632*x^5 + 53719*x^4 - 61132*x^3 + 69938*x^2 - 73695*x + 83521)
 
gp: K = bnfinit(x^12 - x^11 + 16*x^10 - 14*x^9 + 237*x^8 - 334*x^7 + 3651*x^6 - 4632*x^5 + 53719*x^4 - 61132*x^3 + 69938*x^2 - 73695*x + 83521, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 16 x^{10} - 14 x^{9} + 237 x^{8} - 334 x^{7} + 3651 x^{6} - 4632 x^{5} + 53719 x^{4} - 61132 x^{3} + 69938 x^{2} - 73695 x + 83521 \)

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:  \(39155492640500000000=2^{8}\cdot 5^{9}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.93$
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:  $2, 5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{201884435} a^{9} + \frac{10348681}{201884435} a^{8} - \frac{76621279}{201884435} a^{7} + \frac{29967076}{201884435} a^{6} + \frac{6063491}{201884435} a^{5} - \frac{50729422}{201884435} a^{4} - \frac{76338487}{201884435} a^{3} + \frac{73021333}{201884435} a^{2} - \frac{61654462}{201884435} a + \frac{3594389}{11875555}$, $\frac{1}{3432035395} a^{10} - \frac{1}{3432035395} a^{9} + \frac{6277096}{3432035395} a^{8} + \frac{326938819}{686407079} a^{7} - \frac{1540537672}{3432035395} a^{6} + \frac{731072851}{3432035395} a^{5} + \frac{518626289}{3432035395} a^{4} - \frac{1479359218}{3432035395} a^{3} - \frac{1471961712}{3432035395} a^{2} - \frac{18890276}{40376887} a - \frac{3667666}{11875555}$, $\frac{1}{291723008575} a^{11} - \frac{7}{58344601715} a^{10} - \frac{239}{291723008575} a^{9} - \frac{20094283268}{291723008575} a^{8} - \frac{15062574091}{291723008575} a^{7} - \frac{2724550113}{11668920343} a^{6} - \frac{42995932249}{291723008575} a^{5} - \frac{126472895436}{291723008575} a^{4} - \frac{57521142507}{291723008575} a^{3} - \frac{2958397552}{17160176975} a^{2} - \frac{192675359}{1009422175} a + \frac{27261543}{59377775}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( \frac{3317}{3432035395} a^{11} - \frac{53072}{3432035395} a^{10} + \frac{46438}{3432035395} a^{9} - \frac{786129}{3432035395} a^{8} + \frac{584232}{3432035395} a^{7} - \frac{12110367}{3432035395} a^{6} + \frac{15364344}{3432035395} a^{5} - \frac{178185923}{3432035395} a^{4} + \frac{11927932}{201884435} a^{3} - \frac{2845696712}{3432035395} a^{2} + \frac{169167}{2375111} a - \frac{958613}{11875555} \) (order $10$)
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:  \( 14744.5413899 \) (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

\(\Q(\sqrt{5}) \), 3.3.10580.1 x3, \(\Q(\zeta_{5})\), 6.6.559682000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$2$2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$23$23.12.8.1$x^{12} - 69 x^{9} + 1587 x^{6} - 12167 x^{3} + 372468371$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$