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Properties

Label 14.0.61820496347...0639.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,6719^{7}$
Root discriminant $81.97$
Ramified prime $6719$
Class number $15$ (GRH)
Class group $[15]$ (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![4742837, -9406012, 6395526, -1200614, 203477, -223603, 87904, 11296, -3200, -1392, 549, 74, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 74*x^11 + 549*x^10 - 1392*x^9 - 3200*x^8 + 11296*x^7 + 87904*x^6 - 223603*x^5 + 203477*x^4 - 1200614*x^3 + 6395526*x^2 - 9406012*x + 4742837)
 
gp: K = bnfinit(x^14 - 2*x^13 + 74*x^11 + 549*x^10 - 1392*x^9 - 3200*x^8 + 11296*x^7 + 87904*x^6 - 223603*x^5 + 203477*x^4 - 1200614*x^3 + 6395526*x^2 - 9406012*x + 4742837, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 74 x^{11} + 549 x^{10} - 1392 x^{9} - 3200 x^{8} + 11296 x^{7} + 87904 x^{6} - 223603 x^{5} + 203477 x^{4} - 1200614 x^{3} + 6395526 x^{2} - 9406012 x + 4742837 \)

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:  \(-618204963477212079917000639=-\,6719^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.97$
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:  $6719$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19} a^{10} + \frac{6}{19} a^{9} - \frac{5}{19} a^{8} - \frac{7}{19} a^{7} + \frac{4}{19} a^{6} + \frac{2}{19} a^{5} + \frac{7}{19} a^{3} - \frac{8}{19} a$, $\frac{1}{19} a^{11} - \frac{3}{19} a^{9} + \frac{4}{19} a^{8} + \frac{8}{19} a^{7} - \frac{3}{19} a^{6} + \frac{7}{19} a^{5} + \frac{7}{19} a^{4} - \frac{4}{19} a^{3} - \frac{8}{19} a^{2} - \frac{9}{19} a$, $\frac{1}{20299961} a^{12} + \frac{410437}{20299961} a^{11} - \frac{239710}{20299961} a^{10} + \frac{370347}{1845451} a^{9} + \frac{446747}{1068419} a^{8} + \frac{9309991}{20299961} a^{7} + \frac{3714346}{20299961} a^{6} + \frac{2649962}{20299961} a^{5} - \frac{3079189}{20299961} a^{4} + \frac{482290}{20299961} a^{3} + \frac{10085655}{20299961} a^{2} + \frac{9242545}{20299961} a - \frac{31089}{97129}$, $\frac{1}{3812413830515554858213707883344197} a^{13} + \frac{70772556679286885009599105}{3812413830515554858213707883344197} a^{12} - \frac{33394953196473128563957921950087}{3812413830515554858213707883344197} a^{11} + \frac{5161655119104643353918157200269}{346583075501414078019427989394927} a^{10} - \frac{233843268899416256247624526601445}{3812413830515554858213707883344197} a^{9} - \frac{1110220581266718542140718642345937}{3812413830515554858213707883344197} a^{8} - \frac{843056575983404299770815002302644}{3812413830515554858213707883344197} a^{7} - \frac{796541261093626680786160532549610}{3812413830515554858213707883344197} a^{6} - \frac{1787948559787743354246780752860525}{3812413830515554858213707883344197} a^{5} - \frac{961209355307697969170256056147270}{3812413830515554858213707883344197} a^{4} - \frac{284262990155146846027637229063081}{3812413830515554858213707883344197} a^{3} + \frac{1621801157500106445934077081129861}{3812413830515554858213707883344197} a^{2} + \frac{165824423483225296640408938645886}{346583075501414078019427989394927} a - \frac{83394160231918577228389240121}{793096282611931528648576634771}$

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

Class group and class number

$C_{15}$, which has order $15$ (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:  \( 5562647.9127 \) (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

\(\Q(\sqrt{-6719}) \), 7.1.303328992959.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.303328992959.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
6719Data not computed