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Properties

Label 12.0.10208309886...744.13
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 7^{10}\cdot 13^{10}$
Root discriminant $121.36$
Ramified primes $2, 7, 13$
Class number $160056$ (GRH)
Class group $[3, 3, 17784]$ (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![125095671, -2851794, 19852557, 312844, 1326358, 29528, 58064, -1182, 1683, -70, 23, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 23*x^10 - 70*x^9 + 1683*x^8 - 1182*x^7 + 58064*x^6 + 29528*x^5 + 1326358*x^4 + 312844*x^3 + 19852557*x^2 - 2851794*x + 125095671)
 
gp: K = bnfinit(x^12 - 2*x^11 + 23*x^10 - 70*x^9 + 1683*x^8 - 1182*x^7 + 58064*x^6 + 29528*x^5 + 1326358*x^4 + 312844*x^3 + 19852557*x^2 - 2851794*x + 125095671, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 23 x^{10} - 70 x^{9} + 1683 x^{8} - 1182 x^{7} + 58064 x^{6} + 29528 x^{5} + 1326358 x^{4} + 312844 x^{3} + 19852557 x^{2} - 2851794 x + 125095671 \)

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:  \(10208309886795316042399744=2^{18}\cdot 7^{10}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.36$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(728=2^{3}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{728}(1,·)$, $\chi_{728}(549,·)$, $\chi_{728}(81,·)$, $\chi_{728}(9,·)$, $\chi_{728}(173,·)$, $\chi_{728}(61,·)$, $\chi_{728}(337,·)$, $\chi_{728}(181,·)$, $\chi_{728}(361,·)$, $\chi_{728}(121,·)$, $\chi_{728}(573,·)$, $\chi_{728}(101,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{189} a^{6} + \frac{2}{63} a^{5} - \frac{17}{189} a^{4} - \frac{1}{63} a^{3} + \frac{43}{189} a^{2} + \frac{8}{63} a - \frac{1}{7}$, $\frac{1}{189} a^{7} + \frac{10}{189} a^{5} - \frac{1}{7} a^{4} - \frac{2}{189} a^{3} + \frac{3}{7} a^{2} + \frac{2}{21} a - \frac{1}{7}$, $\frac{1}{189} a^{8} - \frac{8}{63} a^{5} - \frac{1}{9} a^{4} - \frac{5}{63} a^{3} - \frac{34}{189} a^{2} - \frac{5}{63} a + \frac{3}{7}$, $\frac{1}{5103} a^{9} - \frac{1}{567} a^{8} - \frac{4}{1701} a^{7} + \frac{4}{1701} a^{6} - \frac{113}{1701} a^{5} - \frac{185}{1701} a^{4} + \frac{647}{5103} a^{3} - \frac{383}{1701} a^{2} + \frac{25}{189} a + \frac{2}{7}$, $\frac{1}{5103} a^{10} - \frac{4}{1701} a^{8} + \frac{4}{1701} a^{7} + \frac{4}{1701} a^{6} + \frac{130}{1701} a^{5} + \frac{512}{5103} a^{4} + \frac{271}{1701} a^{3} - \frac{64}{189} a^{2} + \frac{3}{7} a$, $\frac{1}{10092326736718518520347} a^{11} + \frac{164699205066976960}{3364108912239506173449} a^{10} + \frac{298232463579746204}{10092326736718518520347} a^{9} + \frac{2180384829958164439}{3364108912239506173449} a^{8} + \frac{5398053065357580923}{3364108912239506173449} a^{7} + \frac{6299678706596101}{373789879137722908161} a^{6} + \frac{683274035776361969993}{10092326736718518520347} a^{5} - \frac{270080576608852570030}{3364108912239506173449} a^{4} - \frac{934041983629474609637}{10092326736718518520347} a^{3} + \frac{847639499844733716341}{3364108912239506173449} a^{2} + \frac{158635839411778434773}{373789879137722908161} a + \frac{17432093114210263}{137069996016766743}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{17784}$, which has order $160056$ (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:  \( 17569.03784509895 \) (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

\(\Q(\sqrt{-182}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{13}) \), 3.3.8281.2, \(\Q(\sqrt{13}, \sqrt{-14})\), 6.0.3195044582912.4, 6.0.245772660224.3, 6.6.891474493.2

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 ${\href{/LocalNumberField/3.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$7$7.12.10.2$x^{12} + 35 x^{6} + 441$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$