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Properties

Label 14.0.69684751861...0039.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,4919^{7}$
Root discriminant $70.14$
Ramified prime $4919$
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![2602151, 0, -620741, 0, 206223, 0, 127838, 0, 21314, 0, 1633, 0, 62, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 62*x^12 + 1633*x^10 + 21314*x^8 + 127838*x^6 + 206223*x^4 - 620741*x^2 + 2602151)
 
gp: K = bnfinit(x^14 + 62*x^12 + 1633*x^10 + 21314*x^8 + 127838*x^6 + 206223*x^4 - 620741*x^2 + 2602151, 1)
 

Normalized defining polynomial

\( x^{14} + 62 x^{12} + 1633 x^{10} + 21314 x^{8} + 127838 x^{6} + 206223 x^{4} - 620741 x^{2} + 2602151 \)

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:  \(-69684751861829721459380039=-\,4919^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.14$
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:  $4919$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{22} a^{8} + \frac{1}{22} a^{6} - \frac{5}{11} a^{4} - \frac{1}{2} a^{3} + \frac{1}{22} a^{2} - \frac{1}{2} a - \frac{5}{11}$, $\frac{1}{22} a^{9} + \frac{1}{22} a^{7} - \frac{5}{11} a^{5} - \frac{1}{2} a^{4} + \frac{1}{22} a^{3} - \frac{1}{2} a^{2} - \frac{5}{11} a$, $\frac{1}{19778} a^{10} - \frac{168}{9889} a^{8} - \frac{8927}{19778} a^{6} - \frac{1}{2} a^{5} + \frac{2029}{19778} a^{4} - \frac{7607}{19778} a^{2} - \frac{1}{2} a + \frac{3148}{9889}$, $\frac{1}{19778} a^{11} - \frac{168}{9889} a^{9} + \frac{481}{9889} a^{7} - \frac{1}{2} a^{6} - \frac{3930}{9889} a^{5} - \frac{7607}{19778} a^{3} - \frac{3593}{19778} a - \frac{1}{2}$, $\frac{1}{14263900265186} a^{12} - \frac{263788685}{14263900265186} a^{10} - \frac{114624788077}{7131950132593} a^{8} - \frac{3377248198901}{14263900265186} a^{6} + \frac{738896936759}{7131950132593} a^{4} + \frac{566640400199}{14263900265186} a^{2} - \frac{1}{2} a + \frac{1338782262577}{14263900265186}$, $\frac{1}{328069706099278} a^{13} + \frac{1671106500}{164034853049639} a^{11} - \frac{147214932661}{14263900265186} a^{9} - \frac{37513102549785}{328069706099278} a^{7} - \frac{1}{2} a^{6} + \frac{71036845176831}{328069706099278} a^{5} - \frac{1}{2} a^{4} + \frac{85301910395003}{328069706099278} a^{3} - \frac{1}{2} a^{2} + \frac{86284642755785}{328069706099278} a$

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:  \( 4203376.418 \) (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

\(\Q(\sqrt{-4919}) \), 7.1.119022883559.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.119022883559.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
4919Data not computed