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Properties

Label 14.0.54638052730...0751.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,4751^{7}$
Root discriminant $68.93$
Ramified prime $4751$
Class number $13$ (GRH)
Class group $[13]$ (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![1331177, 132111, -544712, -101850, 251350, 36976, -36573, -5588, 6405, -117, -264, 62, 7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 7*x^12 + 62*x^11 - 264*x^10 - 117*x^9 + 6405*x^8 - 5588*x^7 - 36573*x^6 + 36976*x^5 + 251350*x^4 - 101850*x^3 - 544712*x^2 + 132111*x + 1331177)
 
gp: K = bnfinit(x^14 - 6*x^13 + 7*x^12 + 62*x^11 - 264*x^10 - 117*x^9 + 6405*x^8 - 5588*x^7 - 36573*x^6 + 36976*x^5 + 251350*x^4 - 101850*x^3 - 544712*x^2 + 132111*x + 1331177, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} + 7 x^{12} + 62 x^{11} - 264 x^{10} - 117 x^{9} + 6405 x^{8} - 5588 x^{7} - 36573 x^{6} + 36976 x^{5} + 251350 x^{4} - 101850 x^{3} - 544712 x^{2} + 132111 x + 1331177 \)

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:  \(-54638052730056427270720751=-\,4751^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.93$
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:  $4751$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} - \frac{4}{13} a^{11} + \frac{6}{13} a^{10} - \frac{6}{13} a^{9} - \frac{3}{13} a^{7} + \frac{3}{13} a^{6} + \frac{4}{13} a^{4} - \frac{1}{13} a^{3} - \frac{5}{13} a^{2} + \frac{1}{13} a - \frac{6}{13}$, $\frac{1}{40472903198164590954871501416242297209} a^{13} + \frac{172604963270025043743240108015894499}{40472903198164590954871501416242297209} a^{12} - \frac{14282481520948615823570301624568866785}{40472903198164590954871501416242297209} a^{11} + \frac{19520428534394246459625738372976006925}{40472903198164590954871501416242297209} a^{10} + \frac{18447371001010092232455318627017439051}{40472903198164590954871501416242297209} a^{9} + \frac{12554965988687837951512929227039481182}{40472903198164590954871501416242297209} a^{8} + \frac{8391325745527828359470428408858310063}{40472903198164590954871501416242297209} a^{7} - \frac{18302196225650014936457876009975502001}{40472903198164590954871501416242297209} a^{6} + \frac{10558399041016848760790301929097469145}{40472903198164590954871501416242297209} a^{5} + \frac{989093062192047983143192825191791380}{40472903198164590954871501416242297209} a^{4} + \frac{13624182148920241831508751621348455595}{40472903198164590954871501416242297209} a^{3} + \frac{7569526520469180599264223678694343740}{40472903198164590954871501416242297209} a^{2} - \frac{10330084498830546150148876138466797198}{40472903198164590954871501416242297209} a - \frac{4921146225563117745190923565348412693}{40472903198164590954871501416242297209}$

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

Class group and class number

$C_{13}$, which has order $13$ (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:  \( 461238.688486 \) (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

\(\Q(\sqrt{-4751}) \), 7.1.107239576751.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.107239576751.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
4751Data not computed