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Properties

Label 12.0.269254866892578125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 13^{10}$
Root discriminant $28.35$
Ramified primes $5, 13$
Class number $8$
Class group $[2, 2, 2]$
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![521, 309, 415, 505, 471, 155, 24, -55, 1, -15, 10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 10*x^10 - 15*x^9 + x^8 - 55*x^7 + 24*x^6 + 155*x^5 + 471*x^4 + 505*x^3 + 415*x^2 + 309*x + 521)
 
gp: K = bnfinit(x^12 - x^11 + 10*x^10 - 15*x^9 + x^8 - 55*x^7 + 24*x^6 + 155*x^5 + 471*x^4 + 505*x^3 + 415*x^2 + 309*x + 521, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 10 x^{10} - 15 x^{9} + x^{8} - 55 x^{7} + 24 x^{6} + 155 x^{5} + 471 x^{4} + 505 x^{3} + 415 x^{2} + 309 x + 521 \)

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:  \(269254866892578125=5^{9}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.35$
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:  $5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(65=5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{65}(1,·)$, $\chi_{65}(38,·)$, $\chi_{65}(9,·)$, $\chi_{65}(43,·)$, $\chi_{65}(12,·)$, $\chi_{65}(14,·)$, $\chi_{65}(29,·)$, $\chi_{65}(16,·)$, $\chi_{65}(17,·)$, $\chi_{65}(23,·)$, $\chi_{65}(61,·)$, $\chi_{65}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{85801} a^{10} - \frac{3219}{85801} a^{9} - \frac{7566}{85801} a^{8} + \frac{33218}{85801} a^{7} + \frac{22401}{85801} a^{6} - \frac{37020}{85801} a^{5} + \frac{8125}{85801} a^{4} + \frac{12482}{85801} a^{3} + \frac{34561}{85801} a^{2} - \frac{30321}{85801} a + \frac{13654}{85801}$, $\frac{1}{130958838509} a^{11} + \frac{611461}{130958838509} a^{10} + \frac{1801557972}{130958838509} a^{9} - \frac{53911649593}{130958838509} a^{8} - \frac{3827098138}{130958838509} a^{7} + \frac{290930570}{130958838509} a^{6} + \frac{12217985936}{130958838509} a^{5} + \frac{47872822426}{130958838509} a^{4} - \frac{26200820668}{130958838509} a^{3} + \frac{19955279940}{130958838509} a^{2} - \frac{10907114327}{130958838509} a + \frac{28928194657}{130958838509}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 615.54450504 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.169.1, 4.0.21125.1, 6.6.3570125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$13$13.12.10.5$x^{12} + 65 x^{6} + 1352$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$