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Properties

Label 12.0.58149447531...2209.1
Degree $12$
Signature $[0, 6]$
Discriminant $13^{10}\cdot 59^{6}$
Root discriminant $65.12$
Ramified primes $13, 59$
Class number $4851$ (GRH)
Class group $[7, 693]$ (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![20804159, -3880090, 6437932, -1065793, 887119, -125215, 69264, -7886, 3225, -268, 85, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 85*x^10 - 268*x^9 + 3225*x^8 - 7886*x^7 + 69264*x^6 - 125215*x^5 + 887119*x^4 - 1065793*x^3 + 6437932*x^2 - 3880090*x + 20804159)
 
gp: K = bnfinit(x^12 - 4*x^11 + 85*x^10 - 268*x^9 + 3225*x^8 - 7886*x^7 + 69264*x^6 - 125215*x^5 + 887119*x^4 - 1065793*x^3 + 6437932*x^2 - 3880090*x + 20804159, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 85 x^{10} - 268 x^{9} + 3225 x^{8} - 7886 x^{7} + 69264 x^{6} - 125215 x^{5} + 887119 x^{4} - 1065793 x^{3} + 6437932 x^{2} - 3880090 x + 20804159 \)

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:  \(5814944753134268792209=13^{10}\cdot 59^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.12$
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:  $13, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(767=13\cdot 59\)
Dirichlet character group:    $\lbrace$$\chi_{767}(1,·)$, $\chi_{767}(355,·)$, $\chi_{767}(296,·)$, $\chi_{767}(235,·)$, $\chi_{767}(237,·)$, $\chi_{767}(589,·)$, $\chi_{767}(178,·)$, $\chi_{767}(532,·)$, $\chi_{767}(471,·)$, $\chi_{767}(530,·)$, $\chi_{767}(412,·)$, $\chi_{767}(766,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{114054418297861509695415801} a^{11} - \frac{16937944433829621907659595}{114054418297861509695415801} a^{10} - \frac{8316502328534387580048682}{114054418297861509695415801} a^{9} - \frac{44415361059321801101009815}{114054418297861509695415801} a^{8} + \frac{20761855422731229457392298}{114054418297861509695415801} a^{7} - \frac{1901733841377855795416782}{114054418297861509695415801} a^{6} - \frac{9146109627871622382555047}{38018139432620503231805267} a^{5} - \frac{55178793598210211935307110}{114054418297861509695415801} a^{4} + \frac{18098738958880129505135111}{38018139432620503231805267} a^{3} - \frac{4923928839605041242560966}{114054418297861509695415801} a^{2} - \frac{18653672321583200324840634}{38018139432620503231805267} a + \frac{1389981568106229951033487}{38018139432620503231805267}$

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

Class group and class number

$C_{7}\times C_{693}$, which has order $4851$ (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:  \( 120.78403136265631 \) (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

\(\Q(\sqrt{-767}) \), \(\Q(\sqrt{-59}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\sqrt{13}, \sqrt{-59})\), 6.0.76255785047.1, 6.0.5865829619.2, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ R

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
$13$13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$59$59.12.6.1$x^{12} + 9447434 x^{6} - 714924299 x^{2} + 22313502296089$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$