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Properties

Label 12.0.21303969908...0625.2
Degree $12$
Signature $[0, 6]$
Discriminant $5^{6}\cdot 7^{10}\cdot 13^{6}$
Root discriminant $40.80$
Ramified primes $5, 7, 13$
Class number $320$ (GRH)
Class group $[4, 4, 20]$ (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![271181, -70165, 141674, -9786, 36700, -4588, 6649, -1200, 748, -112, 43, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 43*x^10 - 112*x^9 + 748*x^8 - 1200*x^7 + 6649*x^6 - 4588*x^5 + 36700*x^4 - 9786*x^3 + 141674*x^2 - 70165*x + 271181)
 
gp: K = bnfinit(x^12 - 4*x^11 + 43*x^10 - 112*x^9 + 748*x^8 - 1200*x^7 + 6649*x^6 - 4588*x^5 + 36700*x^4 - 9786*x^3 + 141674*x^2 - 70165*x + 271181, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 43 x^{10} - 112 x^{9} + 748 x^{8} - 1200 x^{7} + 6649 x^{6} - 4588 x^{5} + 36700 x^{4} - 9786 x^{3} + 141674 x^{2} - 70165 x + 271181 \)

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:  \(21303969908600640625=5^{6}\cdot 7^{10}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.80$
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:  $5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(455=5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{455}(1,·)$, $\chi_{455}(194,·)$, $\chi_{455}(261,·)$, $\chi_{455}(326,·)$, $\chi_{455}(129,·)$, $\chi_{455}(454,·)$, $\chi_{455}(79,·)$, $\chi_{455}(144,·)$, $\chi_{455}(274,·)$, $\chi_{455}(181,·)$, $\chi_{455}(311,·)$, $\chi_{455}(376,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{97} a^{8} + \frac{48}{97} a^{7} + \frac{35}{97} a^{6} + \frac{48}{97} a^{5} + \frac{39}{97} a^{4} + \frac{48}{97} a^{3} + \frac{39}{97} a^{2} + \frac{47}{97} a + \frac{47}{97}$, $\frac{1}{97} a^{9} - \frac{38}{97} a^{7} + \frac{17}{97} a^{6} - \frac{34}{97} a^{5} + \frac{19}{97} a^{4} - \frac{34}{97} a^{3} + \frac{18}{97} a^{2} + \frac{22}{97} a - \frac{25}{97}$, $\frac{1}{97} a^{10} - \frac{2}{97} a^{7} + \frac{35}{97} a^{6} - \frac{7}{97} a^{4} - \frac{1}{97} a^{3} - \frac{48}{97} a^{2} + \frac{15}{97} a + \frac{40}{97}$, $\frac{1}{230761303962880602913} a^{11} + \frac{30204860788875331}{7957286343547606997} a^{10} - \frac{401043796746820}{230761303962880602913} a^{9} - \frac{162986191819915992}{230761303962880602913} a^{8} + \frac{113172032434501536627}{230761303962880602913} a^{7} + \frac{56764619086138693904}{230761303962880602913} a^{6} + \frac{98381470424924581220}{230761303962880602913} a^{5} - \frac{43979207525826446939}{230761303962880602913} a^{4} + \frac{7371761984188671292}{230761303962880602913} a^{3} + \frac{25525384821045866394}{230761303962880602913} a^{2} + \frac{92279319557369556711}{230761303962880602913} a - \frac{18977066502213760369}{230761303962880602913}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{20}$, which has order $320$ (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

\(\Q(\sqrt{-455}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-91}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{5}, \sqrt{-91})\), 6.0.4615622375.1, 6.6.300125.1, 6.0.36924979.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$