↑

Properties

Label 14.0.30192443725...0091.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,1171^{7}$
Root discriminant $34.22$
Ramified prime $1171$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{7}$ (as 14T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50256, 21780, 18961, 8688, 1208, -5028, -2544, -780, 310, 336, 96, -36, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 16*x^12 - 36*x^11 + 96*x^10 + 336*x^9 + 310*x^8 - 780*x^7 - 2544*x^6 - 5028*x^5 + 1208*x^4 + 8688*x^3 + 18961*x^2 + 21780*x + 50256)
 
gp: K = bnfinit(x^14 - 16*x^12 - 36*x^11 + 96*x^10 + 336*x^9 + 310*x^8 - 780*x^7 - 2544*x^6 - 5028*x^5 + 1208*x^4 + 8688*x^3 + 18961*x^2 + 21780*x + 50256, 1)
 

Normalized defining polynomial

\( x^{14} - 16 x^{12} - 36 x^{11} + 96 x^{10} + 336 x^{9} + 310 x^{8} - 780 x^{7} - 2544 x^{6} - 5028 x^{5} + 1208 x^{4} + 8688 x^{3} + 18961 x^{2} + 21780 x + 50256 \)

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:  \(-3019244372533000260091=-\,1171^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.22$
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:  $1171$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{7} - \frac{1}{6} a^{5} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{5}{12} a$, $\frac{1}{24} a^{8} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} + \frac{1}{6} a^{4} - \frac{1}{4} a^{3} - \frac{11}{24} a^{2}$, $\frac{1}{24} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{5}{12} a$, $\frac{1}{24} a^{10} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{5}{12} a^{2} - \frac{1}{4} a$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{8} - \frac{1}{12} a^{6} - \frac{3}{16} a^{5} + \frac{1}{6} a^{4} + \frac{5}{12} a^{3} + \frac{17}{48} a^{2}$, $\frac{1}{672} a^{12} - \frac{1}{224} a^{11} - \frac{1}{56} a^{10} - \frac{13}{672} a^{9} - \frac{1}{672} a^{8} - \frac{1}{42} a^{7} - \frac{85}{672} a^{6} + \frac{59}{672} a^{5} - \frac{1}{12} a^{4} + \frac{137}{672} a^{3} - \frac{295}{672} a^{2} + \frac{1}{56} a + \frac{5}{14}$, $\frac{1}{30099723383004576} a^{13} - \frac{931809613611}{10033241127668192} a^{12} - \frac{30811541216161}{7524930845751144} a^{11} - \frac{335522282341657}{30099723383004576} a^{10} + \frac{1896249843505}{30099723383004576} a^{9} + \frac{83515872086}{313538785239631} a^{8} + \frac{536462827052059}{30099723383004576} a^{7} + \frac{1262689047406187}{10033241127668192} a^{6} + \frac{1803544587373}{1254155140958524} a^{5} + \frac{1852081869309951}{10033241127668192} a^{4} - \frac{4686817519096705}{30099723383004576} a^{3} + \frac{876488683851601}{7524930845751144} a^{2} - \frac{300514517234955}{1254155140958524} a - \frac{58165991078255}{313538785239631}$

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:  \( 657151.249352 \) (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

\(\Q(\sqrt{-1171}) \), 7.1.1605723211.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.1605723211.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
1171Data not computed