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Properties

Label 13.13.3028751065...0000.1
Degree $13$
Signature $[13, 0]$
Discriminant $2^{12}\cdot 5^{12}\cdot 13^{13}$
Root discriminant $108.90$
Ramified primes $2, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_{13}$ (as 13T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-69500, 203125, 0, -284375, 0, 113750, 0, -19500, 0, 1625, 0, -65, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 65*x^11 + 1625*x^9 - 19500*x^7 + 113750*x^5 - 284375*x^3 + 203125*x - 69500)
 
gp: K = bnfinit(x^13 - 65*x^11 + 1625*x^9 - 19500*x^7 + 113750*x^5 - 284375*x^3 + 203125*x - 69500, 1)
 

Normalized defining polynomial

\( x^{13} - 65 x^{11} + 1625 x^{9} - 19500 x^{7} + 113750 x^{5} - 284375 x^{3} + 203125 x - 69500 \)

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

Invariants

Degree:  $13$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[13, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(302875106592253000000000000=2^{12}\cdot 5^{12}\cdot 13^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $108.90$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{145} a^{7} + \frac{11}{145} a^{6} - \frac{6}{145} a^{5} - \frac{8}{29} a^{4} + \frac{12}{29} a^{3} + \frac{2}{29} a^{2} - \frac{1}{29} a + \frac{1}{29}$, $\frac{1}{145} a^{8} - \frac{11}{145} a^{6} - \frac{3}{145} a^{5} + \frac{13}{29} a^{4} - \frac{14}{29} a^{3} + \frac{6}{29} a^{2} + \frac{12}{29} a - \frac{11}{29}$, $\frac{1}{725} a^{9} + \frac{12}{145} a^{6} - \frac{6}{145} a^{5} - \frac{3}{29} a^{4} - \frac{13}{29} a^{3} + \frac{1}{29} a^{2} + \frac{13}{29} a + \frac{8}{29}$, $\frac{1}{725} a^{10} + \frac{7}{145} a^{6} - \frac{1}{145} a^{5} - \frac{4}{29} a^{4} + \frac{2}{29} a^{3} - \frac{11}{29} a^{2} - \frac{9}{29} a - \frac{12}{29}$, $\frac{1}{725} a^{11} + \frac{9}{145} a^{6} - \frac{7}{145} a^{5} - \frac{8}{29} a^{3} + \frac{6}{29} a^{2} - \frac{5}{29} a - \frac{7}{29}$, $\frac{1}{725} a^{12} + \frac{2}{29} a^{6} - \frac{4}{145} a^{5} + \frac{6}{29} a^{4} + \frac{14}{29} a^{3} + \frac{6}{29} a^{2} + \frac{2}{29} a - \frac{9}{29}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $12$
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:  \( 4111386187.29 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_{13}$ (as 13T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 156
The 13 conjugacy class representatives for $F_{13}$
Character table for $F_{13}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 26 sibling: data not computed
Degree 39 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
$5$5.13.12.1$x^{13} - 5$$13$$1$$12$$C_{13}:C_4$$[\ ]_{13}^{4}$
$13$13.13.13.1$x^{13} + 13 x + 13$$13$$1$$13$$F_{13}$$[13/12]_{12}$