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Properties

Label 12.12.3500313269...5625.2
Degree $12$
Signature $[12, 0]$
Discriminant $5^{9}\cdot 13^{11}$
Root discriminant $35.10$
Ramified primes $5, 13$
Class number $2$
Class group $[2]$
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![1, -1, -90, -170, 586, 350, -571, -170, 196, 25, -25, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 25*x^10 + 25*x^9 + 196*x^8 - 170*x^7 - 571*x^6 + 350*x^5 + 586*x^4 - 170*x^3 - 90*x^2 - x + 1)
 
gp: K = bnfinit(x^12 - x^11 - 25*x^10 + 25*x^9 + 196*x^8 - 170*x^7 - 571*x^6 + 350*x^5 + 586*x^4 - 170*x^3 - 90*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} - 25 x^{10} + 25 x^{9} + 196 x^{8} - 170 x^{7} - 571 x^{6} + 350 x^{5} + 586 x^{4} - 170 x^{3} - 90 x^{2} - x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3500313269603515625=5^{9}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.10$
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$$\chi_{65}(32,·)$, $\chi_{65}(1,·)$, $\chi_{65}(2,·)$, $\chi_{65}(4,·)$, $\chi_{65}(33,·)$, $\chi_{65}(8,·)$, $\chi_{65}(64,·)$, $\chi_{65}(16,·)$, $\chi_{65}(49,·)$, $\chi_{65}(57,·)$, $\chi_{65}(61,·)$, $\chi_{65}(63,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{20} a^{10} - \frac{1}{10} a^{9} + \frac{3}{20} a^{8} - \frac{1}{4} a^{7} + \frac{1}{5} a^{6} + \frac{3}{10} a^{5} - \frac{2}{5} a^{4} + \frac{9}{20} a^{3} + \frac{1}{5} a^{2} + \frac{1}{4} a + \frac{9}{20}$, $\frac{1}{9843620} a^{11} - \frac{35169}{1968724} a^{10} - \frac{475351}{9843620} a^{9} - \frac{238626}{2460905} a^{8} - \frac{1239591}{9843620} a^{7} + \frac{610041}{2460905} a^{6} + \frac{152701}{2460905} a^{5} + \frac{1949223}{9843620} a^{4} + \frac{2078127}{9843620} a^{3} - \frac{966917}{9843620} a^{2} - \frac{2132063}{4921810} a - \frac{2015727}{9843620}$

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

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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:  \( 215343.952274 \)
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

\(\Q(\sqrt{65}) \), 3.3.169.1, 4.4.274625.1, 6.6.46411625.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
$13$13.12.11.11$x^{12} + 6656$$12$$1$$11$$C_{12}$$[\ ]_{12}$