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Properties

Label 12.0.13334335014...5625.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{6}\cdot 7^{8}\cdot 23^{6}$
Root discriminant $39.24$
Ramified primes $5, 7, 23$
Class number $351$ (GRH)
Class group $[3, 117]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![250879, -115667, 153377, -54863, 37692, -9889, 4774, -934, 396, -68, 27, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 27*x^10 - 68*x^9 + 396*x^8 - 934*x^7 + 4774*x^6 - 9889*x^5 + 37692*x^4 - 54863*x^3 + 153377*x^2 - 115667*x + 250879)
 
gp: K = bnfinit(x^12 - 4*x^11 + 27*x^10 - 68*x^9 + 396*x^8 - 934*x^7 + 4774*x^6 - 9889*x^5 + 37692*x^4 - 54863*x^3 + 153377*x^2 - 115667*x + 250879, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 27 x^{10} - 68 x^{9} + 396 x^{8} - 934 x^{7} + 4774 x^{6} - 9889 x^{5} + 37692 x^{4} - 54863 x^{3} + 153377 x^{2} - 115667 x + 250879 \)

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:  \(13334335014735765625=5^{6}\cdot 7^{8}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.24$
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:  $5, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(805=5\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{805}(576,·)$, $\chi_{805}(1,·)$, $\chi_{805}(484,·)$, $\chi_{805}(459,·)$, $\chi_{805}(781,·)$, $\chi_{805}(114,·)$, $\chi_{805}(436,·)$, $\chi_{805}(599,·)$, $\chi_{805}(344,·)$, $\chi_{805}(116,·)$, $\chi_{805}(666,·)$, $\chi_{805}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{29} a^{10} + \frac{7}{29} a^{9} - \frac{7}{29} a^{8} + \frac{6}{29} a^{7} - \frac{8}{29} a^{6} - \frac{6}{29} a^{5} - \frac{1}{29} a^{4} - \frac{12}{29} a^{3} + \frac{3}{29} a$, $\frac{1}{209471550895307189273} a^{11} - \frac{1980913604578206304}{209471550895307189273} a^{10} + \frac{67885422228646478924}{209471550895307189273} a^{9} - \frac{73412374986704575857}{209471550895307189273} a^{8} + \frac{43552002639583641935}{209471550895307189273} a^{7} + \frac{3225014074930210724}{209471550895307189273} a^{6} - \frac{99890337486890096017}{209471550895307189273} a^{5} + \frac{37397897969918168463}{209471550895307189273} a^{4} + \frac{49034662200354491997}{209471550895307189273} a^{3} + \frac{17074554712908399761}{209471550895307189273} a^{2} + \frac{75089506143974493110}{209471550895307189273} a - \frac{2793209590556685701}{7223156927424385837}$

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

Class group and class number

$C_{3}\times C_{117}$, which has order $351$ (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:  \( 104.882003477 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-115}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{5}, \sqrt{-23})\), 6.0.3651620875.3, 6.0.29212967.1, 6.6.300125.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$23$23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$