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Properties

Label 14.0.15775678870...5827.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,1483^{7}$
Root discriminant $38.51$
Ramified prime $1483$
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![60688, -30912, 55524, -19632, 14093, -2574, 1939, 768, -131, 162, 29, -48, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 15*x^12 - 48*x^11 + 29*x^10 + 162*x^9 - 131*x^8 + 768*x^7 + 1939*x^6 - 2574*x^5 + 14093*x^4 - 19632*x^3 + 55524*x^2 - 30912*x + 60688)
 
gp: K = bnfinit(x^14 - 6*x^13 + 15*x^12 - 48*x^11 + 29*x^10 + 162*x^9 - 131*x^8 + 768*x^7 + 1939*x^6 - 2574*x^5 + 14093*x^4 - 19632*x^3 + 55524*x^2 - 30912*x + 60688, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} + 15 x^{12} - 48 x^{11} + 29 x^{10} + 162 x^{9} - 131 x^{8} + 768 x^{7} + 1939 x^{6} - 2574 x^{5} + 14093 x^{4} - 19632 x^{3} + 55524 x^{2} - 30912 x + 60688 \)

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:  \(-15775678870645415885827=-\,1483^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.51$
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:  $1483$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{1}{12} a^{5} - \frac{1}{12} a^{4} - \frac{5}{12} a^{3} + \frac{5}{12} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{24} a^{8} - \frac{1}{4} a^{4} + \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{6}$, $\frac{1}{24} a^{9} - \frac{1}{4} a^{5} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{6} a$, $\frac{1}{24} a^{10} - \frac{1}{12} a^{6} - \frac{5}{24} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{10} - \frac{1}{48} a^{9} - \frac{1}{24} a^{7} + \frac{1}{24} a^{6} - \frac{11}{48} a^{5} + \frac{11}{48} a^{4} - \frac{7}{16} a^{3} + \frac{1}{8} a^{2} + \frac{5}{12} a - \frac{1}{3}$, $\frac{1}{288} a^{12} - \frac{1}{96} a^{11} + \frac{5}{288} a^{10} - \frac{1}{48} a^{9} - \frac{1}{48} a^{8} - \frac{1}{48} a^{7} + \frac{13}{288} a^{6} + \frac{7}{96} a^{5} + \frac{7}{96} a^{4} + \frac{11}{24} a^{3} + \frac{5}{18} a^{2} - \frac{1}{4} a + \frac{2}{9}$, $\frac{1}{41063708828593075680} a^{13} - \frac{8047356982469501}{20531854414296537840} a^{12} + \frac{18948340730409001}{20531854414296537840} a^{11} + \frac{7371987330648589}{746612887792601376} a^{10} - \frac{30195569160497503}{3421975735716089640} a^{9} + \frac{53035319582942009}{3421975735716089640} a^{8} - \frac{355744854172649489}{41063708828593075680} a^{7} - \frac{738280173720805499}{20531854414296537840} a^{6} + \frac{693780158854122227}{6843951471432179280} a^{5} + \frac{501428467232774423}{13687902942864358560} a^{4} + \frac{203389839831583811}{10265927207148268920} a^{3} + \frac{85420679131984033}{466633054870375860} a^{2} - \frac{9562917282339101}{1026592720714826892} a + \frac{234149190537062014}{1283240900893533615}$

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:  \( 1710902.77008 \) (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

\(\Q(\sqrt{-1483}) \), 7.1.3261545587.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.3261545587.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
1483Data not computed