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Properties

Label 14.0.34510571141...4419.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,859^{7}$
Root discriminant $29.31$
Ramified prime $859$
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![26237, -50890, 41785, -21684, 9920, -3956, 1374, -868, 470, -108, 48, -28, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 5*x^12 - 28*x^11 + 48*x^10 - 108*x^9 + 470*x^8 - 868*x^7 + 1374*x^6 - 3956*x^5 + 9920*x^4 - 21684*x^3 + 41785*x^2 - 50890*x + 26237)
 
gp: K = bnfinit(x^14 - 2*x^13 + 5*x^12 - 28*x^11 + 48*x^10 - 108*x^9 + 470*x^8 - 868*x^7 + 1374*x^6 - 3956*x^5 + 9920*x^4 - 21684*x^3 + 41785*x^2 - 50890*x + 26237, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 5 x^{12} - 28 x^{11} + 48 x^{10} - 108 x^{9} + 470 x^{8} - 868 x^{7} + 1374 x^{6} - 3956 x^{5} + 9920 x^{4} - 21684 x^{3} + 41785 x^{2} - 50890 x + 26237 \)

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:  \(-345105711415338434419=-\,859^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.31$
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:  $859$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{12} a - \frac{1}{12}$, $\frac{1}{12} a^{8} + \frac{1}{4} a^{2} - \frac{1}{3}$, $\frac{1}{72} a^{9} + \frac{1}{36} a^{7} + \frac{1}{72} a^{6} - \frac{1}{18} a^{5} + \frac{1}{18} a^{4} - \frac{13}{72} a^{3} + \frac{7}{18} a^{2} + \frac{17}{36} a - \frac{29}{72}$, $\frac{1}{72} a^{10} + \frac{1}{36} a^{8} + \frac{1}{72} a^{7} - \frac{1}{18} a^{6} + \frac{1}{18} a^{5} - \frac{13}{72} a^{4} - \frac{1}{9} a^{3} + \frac{17}{36} a^{2} - \frac{29}{72} a - \frac{1}{2}$, $\frac{1}{144} a^{11} - \frac{1}{144} a^{10} - \frac{1}{144} a^{9} - \frac{1}{144} a^{8} + \frac{1}{144} a^{7} - \frac{7}{144} a^{6} - \frac{29}{144} a^{5} - \frac{19}{144} a^{4} + \frac{7}{48} a^{3} - \frac{17}{48} a^{2} - \frac{61}{144} a + \frac{1}{48}$, $\frac{1}{864} a^{12} - \frac{1}{216} a^{9} - \frac{1}{108} a^{8} - \frac{1}{108} a^{7} - \frac{5}{432} a^{6} - \frac{1}{27} a^{5} - \frac{1}{27} a^{4} + \frac{49}{216} a^{3} + \frac{37}{108} a^{2} + \frac{13}{108} a - \frac{215}{864}$, $\frac{1}{50409546829056} a^{13} + \frac{21599852915}{50409546829056} a^{12} + \frac{4693021043}{4200795569088} a^{11} + \frac{18099505171}{6301193353632} a^{10} - \frac{11721881495}{6301193353632} a^{9} - \frac{106333891939}{4200795569088} a^{8} - \frac{978612015631}{25204773414528} a^{7} + \frac{1490008769731}{25204773414528} a^{6} - \frac{453322976819}{4200795569088} a^{5} - \frac{237203606759}{2100397784544} a^{4} + \frac{1089245299115}{6301193353632} a^{3} + \frac{553842743681}{1400265189696} a^{2} + \frac{19201806725933}{50409546829056} a + \frac{22738329807911}{50409546829056}$

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:  \( 205733.451757 \) (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

\(\Q(\sqrt{-859}) \), 7.1.633839779.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.633839779.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
859Data not computed