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Properties

Label 14.2.15666620380...8016.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{18}\cdot 3^{12}\cdot 13^{18}$
Root discriminant $169.13$
Ramified primes $2, 3, 13$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $\PSL(2,13)$ (as 14T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27648, -82944, -89856, 0, 0, 7488, 9360, 0, 0, 0, -312, 0, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 312*x^10 + 9360*x^6 + 7488*x^5 - 89856*x^2 - 82944*x - 27648)
 
gp: K = bnfinit(x^14 - 4*x^13 - 312*x^10 + 9360*x^6 + 7488*x^5 - 89856*x^2 - 82944*x - 27648, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 312 x^{10} + 9360 x^{6} + 7488 x^{5} - 89856 x^{2} - 82944 x - 27648 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15666620380205597055220158038016=2^{18}\cdot 3^{12}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $169.13$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4}$, $\frac{1}{8} a^{5} - \frac{1}{2} a$, $\frac{1}{48} a^{6} + \frac{1}{24} a^{5} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{96} a^{7} - \frac{1}{24} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{192} a^{8} + \frac{1}{24} a^{5} + \frac{1}{16} a^{4}$, $\frac{1}{384} a^{9} - \frac{1}{96} a^{5} - \frac{1}{8} a^{4}$, $\frac{1}{2304} a^{10} + \frac{1}{1152} a^{9} + \frac{1}{192} a^{6} - \frac{1}{96} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{2304} a^{11} + \frac{1}{1152} a^{9} - \frac{1}{192} a^{7} - \frac{1}{32} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4608} a^{12} - \frac{1}{1152} a^{9} - \frac{1}{384} a^{8} - \frac{1}{96} a^{5} - \frac{1}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{116573184} a^{13} + \frac{9}{249088} a^{12} + \frac{919}{4483584} a^{11} + \frac{1}{249088} a^{10} - \frac{1}{249088} a^{9} - \frac{81}{62272} a^{8} + \frac{1135}{373632} a^{7} + \frac{1919}{186816} a^{6} + \frac{122}{2919} a^{5} + \frac{649}{15568} a^{4} - \frac{919}{7784} a^{3} + \frac{241}{973} a^{2} - \frac{9}{12649}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $7$
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:  \( 45370423158.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(2,13)$ (as 14T30):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1092
The 9 conjugacy class representatives for $\PSL(2,13)$
Character table for $\PSL(2,13)$

Intermediate fields

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

Sibling fields

Degree 28 sibling: data not computed
Degree 42 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.13.0.1}{13} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.13.12.1$x^{13} - 3$$13$$1$$12$$C_{13}:C_3$$[\ ]_{13}^{3}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.13.18.2$x^{13} + 143 x^{6} + 13$$13$$1$$18$$D_{13}$$[3/2]_{2}$