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Properties

Label 14.10.1569388010...1673.1
Degree $14$
Signature $[10, 2]$
Discriminant $2621\cdot 5261\cdot 113813822033$
Root discriminant $19.94$
Ramified primes $2621, 5261, 113813822033$
Class number $1$
Class group Trivial
Galois group 14T63

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25, 13, 106, -28, -121, 24, 20, -8, 31, -3, -9, 4, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 - 3*x^12 + 4*x^11 - 9*x^10 - 3*x^9 + 31*x^8 - 8*x^7 + 20*x^6 + 24*x^5 - 121*x^4 - 28*x^3 + 106*x^2 + 13*x - 25)
 
gp: K = bnfinit(x^14 - x^13 - 3*x^12 + 4*x^11 - 9*x^10 - 3*x^9 + 31*x^8 - 8*x^7 + 20*x^6 + 24*x^5 - 121*x^4 - 28*x^3 + 106*x^2 + 13*x - 25, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} - 3 x^{12} + 4 x^{11} - 9 x^{10} - 3 x^{9} + 31 x^{8} - 8 x^{7} + 20 x^{6} + 24 x^{5} - 121 x^{4} - 28 x^{3} + 106 x^{2} + 13 x - 25 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1569388010932621673=2621\cdot 5261\cdot 113813822033\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.94$
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:  $2621, 5261, 113813822033$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13829} a^{13} - \frac{401}{13829} a^{12} - \frac{5551}{13829} a^{11} - \frac{6065}{13829} a^{10} + \frac{5916}{13829} a^{9} - \frac{1644}{13829} a^{8} - \frac{6161}{13829} a^{7} + \frac{2830}{13829} a^{6} + \frac{1998}{13829} a^{5} + \frac{2906}{13829} a^{4} - \frac{885}{13829} a^{3} - \frac{5582}{13829} a^{2} + \frac{6437}{13829} a - \frac{2593}{13829}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( 13520.1140622 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T63:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 87178291200
The 135 conjugacy class representatives for S14 are not computed
Character table for S14 is not computed

Intermediate fields

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

Sibling fields

Degree 28 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.13.0.1}{13} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
2621Data not computed
5261Data not computed
113813822033Data not computed