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Properties

Label 13.13.9109989259...8125.1
Degree $13$
Signature $[13, 0]$
Discriminant $3^{6}\cdot 5^{12}\cdot 13^{15}$
Root discriminant $141.49$
Ramified primes $3, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{13}:C_4$ (as 13T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-205335, -296595, 197730, 499395, 106470, -213785, -119145, 10985, 21255, 3900, -390, -130, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 130*x^11 - 390*x^10 + 3900*x^9 + 21255*x^8 + 10985*x^7 - 119145*x^6 - 213785*x^5 + 106470*x^4 + 499395*x^3 + 197730*x^2 - 296595*x - 205335)
 
gp: K = bnfinit(x^13 - 130*x^11 - 390*x^10 + 3900*x^9 + 21255*x^8 + 10985*x^7 - 119145*x^6 - 213785*x^5 + 106470*x^4 + 499395*x^3 + 197730*x^2 - 296595*x - 205335, 1)
 

Normalized defining polynomial

\( x^{13} - 130 x^{11} - 390 x^{10} + 3900 x^{9} + 21255 x^{8} + 10985 x^{7} - 119145 x^{6} - 213785 x^{5} + 106470 x^{4} + 499395 x^{3} + 197730 x^{2} - 296595 x - 205335 \)

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:  \(9109989259587930139892578125=3^{6}\cdot 5^{12}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $141.49$
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:  $3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{78} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{78} a^{8} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{936} a^{9} - \frac{1}{312} a^{8} - \frac{1}{312} a^{7} - \frac{1}{24} a^{6} + \frac{1}{6} a^{5} - \frac{5}{12} a^{4} - \frac{1}{18} a^{3} - \frac{1}{2} a^{2} - \frac{7}{24} a - \frac{1}{8}$, $\frac{1}{936} a^{10} + \frac{1}{24} a^{6} - \frac{1}{4} a^{5} - \frac{11}{36} a^{4} - \frac{1}{2} a^{3} + \frac{1}{24} a^{2} - \frac{1}{2} a - \frac{3}{8}$, $\frac{1}{936} a^{11} + \frac{1}{312} a^{7} - \frac{1}{12} a^{6} + \frac{7}{36} a^{5} - \frac{1}{3} a^{4} - \frac{11}{24} a^{3} + \frac{1}{6} a^{2} + \frac{1}{8} a$, $\frac{1}{262292472} a^{12} - \frac{359}{21857706} a^{11} + \frac{613}{2522043} a^{10} - \frac{399}{4857268} a^{9} - \frac{163841}{87430824} a^{8} - \frac{14425}{21857706} a^{7} - \frac{96199}{5044086} a^{6} - \frac{221081}{1681362} a^{5} - \frac{4274669}{20176344} a^{4} - \frac{401792}{840681} a^{3} - \frac{885593}{6725448} a^{2} - \frac{23951}{373636} a + \frac{10245}{373636}$

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:  \( 254843353531 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{13}.C_2$ (as 13T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 52
The 7 conjugacy class representatives for $C_{13}:C_4$
Character table for $C_{13}:C_4$

Intermediate fields

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

Sibling fields

Degree 26 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.13.12.1$x^{13} - 5$$13$$1$$12$$C_{13}:C_4$$[\ ]_{13}^{4}$
$13$13.13.15.2$x^{13} + 117 x^{3} + 13$$13$$1$$15$$C_{13}:C_4$$[5/4]_{4}$