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Properties

Label 14.0.84659470613...8816.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{25}\cdot 3^{12}\cdot 7^{15}$
Root discriminant $71.12$
Ramified primes $2, 3, 7$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 14T46

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![142814, 0, 295141, 0, -59136, 0, -6517, 0, 742, 0, 231, 0, -28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 28*x^12 + 231*x^10 + 742*x^8 - 6517*x^6 - 59136*x^4 + 295141*x^2 + 142814)
 
gp: K = bnfinit(x^14 - 28*x^12 + 231*x^10 + 742*x^8 - 6517*x^6 - 59136*x^4 + 295141*x^2 + 142814, 1)
 

Normalized defining polynomial

\( x^{14} - 28 x^{12} + 231 x^{10} + 742 x^{8} - 6517 x^{6} - 59136 x^{4} + 295141 x^{2} + 142814 \)

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:  \(-84659470613851429002018816=-\,2^{25}\cdot 3^{12}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.12$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{48} a^{8} + \frac{1}{48} a^{6} - \frac{3}{16} a^{4} + \frac{5}{48} a^{2} + \frac{1}{24}$, $\frac{1}{48} a^{9} + \frac{1}{48} a^{7} - \frac{3}{16} a^{5} + \frac{5}{48} a^{3} + \frac{1}{24} a$, $\frac{1}{1632} a^{10} - \frac{1}{204} a^{8} - \frac{19}{272} a^{6} + \frac{379}{816} a^{4} + \frac{37}{96} a^{2} + \frac{13}{272}$, $\frac{1}{1632} a^{11} - \frac{1}{204} a^{9} + \frac{11}{816} a^{7} - \frac{1}{12} a^{6} - \frac{29}{816} a^{5} - \frac{1}{2} a^{4} + \frac{13}{96} a^{3} + \frac{1}{4} a^{2} + \frac{175}{816} a - \frac{1}{6}$, $\frac{1}{23401425888} a^{12} - \frac{115041}{2600158432} a^{10} - \frac{1155367}{229425744} a^{8} + \frac{143227919}{2925178236} a^{6} - \frac{776242521}{2600158432} a^{4} + \frac{2591028931}{7800475296} a^{2} + \frac{4025282441}{11700712944}$, $\frac{1}{4727088029376} a^{13} - \frac{1}{46802851776} a^{12} - \frac{382721363}{1575696009792} a^{11} + \frac{115041}{5200316864} a^{10} - \frac{4244704}{1448250009} a^{9} - \frac{75507}{9559406} a^{8} + \frac{29150755913}{2363544014688} a^{7} + \frac{1133442295}{23401425888} a^{6} - \frac{574449617257}{1575696009792} a^{5} - \frac{1336386205}{5200316864} a^{4} - \frac{459424622455}{1575696009792} a^{3} - \frac{2434605363}{5200316864} a^{2} + \frac{435603785123}{2363544014688} a + \frac{11088138445}{23401425888}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $6$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 91866342.5122 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T46:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5040
The 15 conjugacy class representatives for 2[1/2]S(7)
Character table for 2[1/2]S(7)

Intermediate fields

\(\Q(\sqrt{-14}) \), 7.1.76846444416.1

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

Sibling fields

Degree 7 sibling: 7.1.76846444416.1
Degree 21 sibling: Deg 21
Degree 30 sibling: data not computed
Degree 35 sibling: 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 R R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.6.6.1$x^{6} + 3 x^{5} - 2$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
3.6.6.1$x^{6} + 3 x^{5} - 2$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
7Data not computed