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Properties

Label 14.0.544488334452342784.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 7^{16}$
Root discriminant $18.49$
Ramified primes $2, 7$
Class number $1$
Class group Trivial
Galois group $F_7$ (as 14T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, -14, 14, 0, 98, -196, 98, 0, 14, -14, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 14*x^11 + 14*x^10 + 98*x^8 - 196*x^7 + 98*x^6 + 14*x^4 - 14*x^3 + 1)
 
gp: K = bnfinit(x^14 - 14*x^11 + 14*x^10 + 98*x^8 - 196*x^7 + 98*x^6 + 14*x^4 - 14*x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{14} - 14 x^{11} + 14 x^{10} + 98 x^{8} - 196 x^{7} + 98 x^{6} + 14 x^{4} - 14 x^{3} + 1 \)

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:  \(-544488334452342784=-\,2^{14}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.49$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{19837} a^{12} + \frac{3865}{19837} a^{11} + \frac{963}{19837} a^{10} + \frac{8597}{19837} a^{9} - \frac{519}{19837} a^{8} + \frac{8842}{19837} a^{7} - \frac{4204}{19837} a^{6} + \frac{8842}{19837} a^{5} - \frac{519}{19837} a^{4} + \frac{8597}{19837} a^{3} + \frac{963}{19837} a^{2} + \frac{3865}{19837} a + \frac{1}{19837}$, $\frac{1}{297555} a^{13} - \frac{1}{297555} a^{12} + \frac{115156}{297555} a^{11} + \frac{7612}{19837} a^{10} + \frac{89639}{297555} a^{9} - \frac{67589}{297555} a^{8} + \frac{110797}{297555} a^{7} - \frac{143693}{297555} a^{6} - \frac{143399}{297555} a^{5} + \frac{110699}{297555} a^{4} - \frac{4505}{19837} a^{3} + \frac{89611}{297555} a^{2} + \frac{114194}{297555} a + \frac{115156}{297555}$

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:  $6$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{392}{3585} a^{13} + \frac{343}{3585} a^{12} + \frac{287}{3585} a^{11} - \frac{385}{239} a^{10} + \frac{343}{3585} a^{9} + \frac{392}{3585} a^{8} + \frac{46019}{3585} a^{7} - \frac{42826}{3585} a^{6} - \frac{343}{3585} a^{5} - \frac{50477}{3585} a^{4} + \frac{3731}{239} a^{3} - \frac{343}{3585} a^{2} - \frac{392}{3585} a - \frac{4018}{3585} \) (order $4$)
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:  \( 6270.5100806 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 14T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\sqrt{-1}) \), 7.1.368947264.1

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

Sibling fields

Galois closure: data not computed
Degree 7 sibling: 7.1.368947264.1
Degree 21 sibling: Deg 21

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
7Data not computed