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Properties

Label 14.0.23075226863...8771.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,811^{7}$
Root discriminant $28.48$
Ramified prime $811$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3568, -1008, -5380, 144, 2641, -216, 542, 270, 101, -36, -35, -18, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 2*x^12 - 18*x^11 - 35*x^10 - 36*x^9 + 101*x^8 + 270*x^7 + 542*x^6 - 216*x^5 + 2641*x^4 + 144*x^3 - 5380*x^2 - 1008*x + 3568)
 
gp: K = bnfinit(x^14 + 2*x^12 - 18*x^11 - 35*x^10 - 36*x^9 + 101*x^8 + 270*x^7 + 542*x^6 - 216*x^5 + 2641*x^4 + 144*x^3 - 5380*x^2 - 1008*x + 3568, 1)
 

Normalized defining polynomial

\( x^{14} + 2 x^{12} - 18 x^{11} - 35 x^{10} - 36 x^{9} + 101 x^{8} + 270 x^{7} + 542 x^{6} - 216 x^{5} + 2641 x^{4} + 144 x^{3} - 5380 x^{2} - 1008 x + 3568 \)

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:  \(-230752268637185668771=-\,811^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.48$
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:  $811$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{12} a^{8} + \frac{1}{4} a^{2} - \frac{1}{3}$, $\frac{1}{24} a^{9} - \frac{1}{24} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{3}{8} a^{2} - \frac{5}{12} a + \frac{1}{6}$, $\frac{1}{48} a^{10} - \frac{1}{48} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{16} a^{4} + \frac{1}{16} a^{3} - \frac{1}{12} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{9} - \frac{1}{24} a^{8} - \frac{1}{8} a^{6} + \frac{3}{16} a^{5} - \frac{1}{4} a^{4} + \frac{23}{48} a^{3} - \frac{1}{4} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{96} a^{12} - \frac{1}{96} a^{10} - \frac{1}{48} a^{9} - \frac{1}{16} a^{7} - \frac{1}{32} a^{6} - \frac{1}{4} a^{5} + \frac{23}{96} a^{4} + \frac{1}{4} a^{3} + \frac{1}{24} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{16743037837807488} a^{13} + \frac{71309368433107}{16743037837807488} a^{12} + \frac{81562002423947}{16743037837807488} a^{11} - \frac{22360480071099}{5581012612602496} a^{10} - \frac{25910364579121}{2790506306301248} a^{9} - \frac{100846282833937}{2790506306301248} a^{8} - \frac{316559629256783}{5581012612602496} a^{7} + \frac{186737379180413}{5581012612602496} a^{6} + \frac{2577612715173683}{16743037837807488} a^{5} + \frac{4169438003416193}{16743037837807488} a^{4} - \frac{726611340385055}{4185759459451872} a^{3} + \frac{337712991703225}{1395253153150624} a^{2} - \frac{20356049018989}{174406644143828} a + \frac{359109737055}{1564185149272}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 166968.173712 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-811}) \), 7.1.533411731.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.533411731.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
811Data not computed