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Properties

Label 14.0.16043017139...0192.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 997^{7}$
Root discriminant $63.15$
Ramified primes $2, 997$
Class number $338$ (GRH)
Class group $[13, 26]$ (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![270304, -384880, 745620, -710096, 237656, 15828, -14255, -376, -536, 672, 122, -56, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 56*x^11 + 122*x^10 + 672*x^9 - 536*x^8 - 376*x^7 - 14255*x^6 + 15828*x^5 + 237656*x^4 - 710096*x^3 + 745620*x^2 - 384880*x + 270304)
 
gp: K = bnfinit(x^14 - 4*x^13 - 56*x^11 + 122*x^10 + 672*x^9 - 536*x^8 - 376*x^7 - 14255*x^6 + 15828*x^5 + 237656*x^4 - 710096*x^3 + 745620*x^2 - 384880*x + 270304, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 56 x^{11} + 122 x^{10} + 672 x^{9} - 536 x^{8} - 376 x^{7} - 14255 x^{6} + 15828 x^{5} + 237656 x^{4} - 710096 x^{3} + 745620 x^{2} - 384880 x + 270304 \)

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:  \(-16043017139485116019720192=-\,2^{14}\cdot 997^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.15$
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:  $2, 997$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{7} + \frac{1}{12} a^{6} + \frac{1}{12} a^{5} + \frac{1}{12} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{24} a^{8} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{3}$, $\frac{1}{240} a^{9} + \frac{1}{120} a^{7} - \frac{7}{60} a^{6} - \frac{13}{240} a^{5} + \frac{7}{30} a^{4} + \frac{49}{120} a^{3} + \frac{17}{60} a^{2} + \frac{11}{30} a - \frac{7}{15}$, $\frac{1}{240} a^{10} + \frac{1}{120} a^{8} - \frac{1}{30} a^{7} + \frac{7}{240} a^{6} - \frac{11}{60} a^{5} - \frac{1}{120} a^{4} + \frac{7}{60} a^{3} + \frac{1}{5} a^{2} - \frac{2}{15} a + \frac{1}{3}$, $\frac{1}{240} a^{11} + \frac{1}{120} a^{8} + \frac{1}{80} a^{7} + \frac{1}{20} a^{6} - \frac{3}{20} a^{5} + \frac{1}{40} a^{4} + \frac{2}{15} a^{3} + \frac{1}{20} a^{2} - \frac{2}{5} a + \frac{4}{15}$, $\frac{1}{2400} a^{12} - \frac{1}{1200} a^{11} - \frac{1}{480} a^{10} - \frac{1}{600} a^{9} - \frac{1}{2400} a^{8} - \frac{23}{1200} a^{7} - \frac{247}{2400} a^{6} - \frac{31}{600} a^{5} - \frac{13}{75} a^{4} + \frac{1}{6} a^{3} + \frac{163}{600} a^{2} + \frac{21}{50} a + \frac{4}{25}$, $\frac{1}{944110585973087762784000} a^{13} - \frac{150285896305053172223}{944110585973087762784000} a^{12} + \frac{1834253223165864964037}{944110585973087762784000} a^{11} - \frac{1250469519027867902159}{944110585973087762784000} a^{10} - \frac{1402509453419552078257}{944110585973087762784000} a^{9} - \frac{12674552961135325249}{188822117194617552556800} a^{8} + \frac{4112961148704544146319}{944110585973087762784000} a^{7} + \frac{60135622811063429414563}{944110585973087762784000} a^{6} + \frac{1362004866909177920913}{9834485270552997529000} a^{5} - \frac{14046916259705712339357}{78675882164423980232000} a^{4} - \frac{116651264371326848248237}{236027646493271940696000} a^{3} - \frac{39117481672943546016207}{78675882164423980232000} a^{2} - \frac{24709378739088051828499}{59006911623317985174000} a + \frac{3376265903371094983271}{9834485270552997529000}$

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

Class group and class number

$C_{13}\times C_{26}$, which has order $338$ (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:  \( 2568027.22257 \) (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

\(\Q(\sqrt{-997}) \), 7.1.63425726272.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.63425726272.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

In the table, R denotes a ramified prime. 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
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
997Data not computed