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Properties

Label 14.0.12297557417...2987.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{15}\cdot 29^{4}\cdot 59^{4}$
Root discriminant $27.23$
Ramified primes $3, 29, 59$
Class number $2$
Class group $[2]$
Galois group $A_7\times C_2$ (as 14T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![361, 494, 2130, 2263, 2826, 1357, 579, -93, -69, 25, 29, 3, -2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 - 2*x^12 + 3*x^11 + 29*x^10 + 25*x^9 - 69*x^8 - 93*x^7 + 579*x^6 + 1357*x^5 + 2826*x^4 + 2263*x^3 + 2130*x^2 + 494*x + 361)
 
gp: K = bnfinit(x^14 - 3*x^13 - 2*x^12 + 3*x^11 + 29*x^10 + 25*x^9 - 69*x^8 - 93*x^7 + 579*x^6 + 1357*x^5 + 2826*x^4 + 2263*x^3 + 2130*x^2 + 494*x + 361, 1)
 

Normalized defining polynomial

\( x^{14} - 3 x^{13} - 2 x^{12} + 3 x^{11} + 29 x^{10} + 25 x^{9} - 69 x^{8} - 93 x^{7} + 579 x^{6} + 1357 x^{5} + 2826 x^{4} + 2263 x^{3} + 2130 x^{2} + 494 x + 361 \)

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:  \(-122975574173606802987=-\,3^{15}\cdot 29^{4}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.23$
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:  $3, 29, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{621} a^{12} + \frac{58}{621} a^{11} + \frac{100}{621} a^{10} - \frac{2}{23} a^{9} + \frac{274}{621} a^{8} - \frac{163}{621} a^{7} - \frac{104}{621} a^{6} - \frac{301}{621} a^{5} - \frac{125}{621} a^{4} + \frac{71}{207} a^{3} + \frac{62}{621} a^{2} + \frac{14}{69} a - \frac{149}{621}$, $\frac{1}{327772908752155407} a^{13} - \frac{2641125889932}{12139737361190941} a^{12} - \frac{3937939794050039}{109257636250718469} a^{11} + \frac{53952380699548082}{327772908752155407} a^{10} - \frac{12326568139871411}{327772908752155407} a^{9} + \frac{12758807671648318}{327772908752155407} a^{8} + \frac{124739883681386975}{327772908752155407} a^{7} + \frac{146752316360150704}{327772908752155407} a^{6} + \frac{4695153790678118}{327772908752155407} a^{5} - \frac{50746016965933600}{327772908752155407} a^{4} - \frac{9717479479523815}{327772908752155407} a^{3} - \frac{5857784105340794}{327772908752155407} a^{2} - \frac{1048966547366471}{327772908752155407} a - \frac{1951038185743348}{17251205723797653}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{289747589}{226355158881} a^{13} - \frac{1706928434}{226355158881} a^{12} + \frac{1775566918}{226355158881} a^{11} + \frac{4265274536}{226355158881} a^{10} + \frac{1729958672}{226355158881} a^{9} - \frac{2100004744}{25150573209} a^{8} - \frac{11549512310}{75451719627} a^{7} + \frac{18238379812}{75451719627} a^{6} + \frac{9587784963}{8383524403} a^{5} - \frac{194454785878}{226355158881} a^{4} - \frac{439722206102}{226355158881} a^{3} - \frac{1055832685253}{226355158881} a^{2} - \frac{532216873942}{226355158881} a - \frac{140104683481}{226355158881} \) (order $6$)
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:  \( 22888.2323492 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_7\times C_2$ (as 14T47):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5040
The 18 conjugacy class representatives for $A_7\times C_2$
Character table for $A_7\times C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 7.7.2134162809.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 42 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ R

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.11.8$x^{6} + 12$$6$$1$$11$$S_3$$[5/2]_{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$59$59.4.0.1$x^{4} - x + 14$$1$$4$$0$$C_4$$[\ ]^{4}$
59.4.0.1$x^{4} - x + 14$$1$$4$$0$$C_4$$[\ ]^{4}$
59.6.4.1$x^{6} + 295 x^{3} + 27848$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$