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Properties

Label 14.0.93021317572...5344.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{28}\cdot 809^{5}$
Root discriminant $43.71$
Ramified primes $2, 809$
Class number $64$
Class group $[2, 32]$
Galois group $C_2\times \PSL(2,7)$ (as 14T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![356, 232, 688, -104, 640, -84, 526, -328, 242, -92, 62, -24, 10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 10*x^12 - 24*x^11 + 62*x^10 - 92*x^9 + 242*x^8 - 328*x^7 + 526*x^6 - 84*x^5 + 640*x^4 - 104*x^3 + 688*x^2 + 232*x + 356)
 
gp: K = bnfinit(x^14 - 2*x^13 + 10*x^12 - 24*x^11 + 62*x^10 - 92*x^9 + 242*x^8 - 328*x^7 + 526*x^6 - 84*x^5 + 640*x^4 - 104*x^3 + 688*x^2 + 232*x + 356, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 10 x^{12} - 24 x^{11} + 62 x^{10} - 92 x^{9} + 242 x^{8} - 328 x^{7} + 526 x^{6} - 84 x^{5} + 640 x^{4} - 104 x^{3} + 688 x^{2} + 232 x + 356 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-93021317572518786105344=-\,2^{28}\cdot 809^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.71$
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:  $2, 809$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{41745228215420798} a^{13} - \frac{8854810261802457}{41745228215420798} a^{12} - \frac{3652791886856726}{20872614107710399} a^{11} - \frac{9939234646935427}{41745228215420798} a^{10} + \frac{1301272820519999}{41745228215420798} a^{9} - \frac{5610584509813887}{41745228215420798} a^{8} + \frac{4822878319179705}{20872614107710399} a^{7} + \frac{6582386529759616}{20872614107710399} a^{6} + \frac{3552750007297863}{20872614107710399} a^{5} - \frac{9132624570654174}{20872614107710399} a^{4} + \frac{10136040156484674}{20872614107710399} a^{3} + \frac{2260353648464413}{20872614107710399} a^{2} - \frac{6769367019217783}{20872614107710399} a + \frac{4975287487015087}{20872614107710399}$

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

Class group and class number

$C_{2}\times C_{32}$, which has order $64$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $6$
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:  \( 19291.3072044 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times \PSL(2,7)$ (as 14T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 336
The 12 conjugacy class representatives for $C_2\times \PSL(2,7)$
Character table for $C_2\times \PSL(2,7)$

Intermediate fields

7.7.670188544.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 14 siblings: data not computed
Degree 16 sibling: data not computed
Degree 28 siblings: data not computed
Degree 42 siblings: data not computed
Arithmetically equvalently siblings: 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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.6.10.5$x^{6} + 2 x^{5} + 6$$6$$1$$10$$S_4\times C_2$$[2, 8/3, 8/3]_{3}^{2}$
2.8.18.65$x^{8} + 6 x^{4} + 4 x^{3} + 6$$8$$1$$18$$S_4\times C_2$$[2, 8/3, 8/3]_{3}^{2}$
809Data not computed