↑

Properties

Label 14.14.4383347556...7968.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{27}\cdot 7^{12}\cdot 223^{3}\cdot 463^{5}$
Root discriminant $575.61$
Ramified primes $2, 7, 223, 463$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T45

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44040163721, 46379325210, 8042614275, -6624859400, -2435588247, 181130646, 160506141, 4488436, -4549640, -240364, 64336, 2720, -457, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 - 457*x^12 + 2720*x^11 + 64336*x^10 - 240364*x^9 - 4549640*x^8 + 4488436*x^7 + 160506141*x^6 + 181130646*x^5 - 2435588247*x^4 - 6624859400*x^3 + 8042614275*x^2 + 46379325210*x + 44040163721)
 
gp: K = bnfinit(x^14 - 6*x^13 - 457*x^12 + 2720*x^11 + 64336*x^10 - 240364*x^9 - 4549640*x^8 + 4488436*x^7 + 160506141*x^6 + 181130646*x^5 - 2435588247*x^4 - 6624859400*x^3 + 8042614275*x^2 + 46379325210*x + 44040163721, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} - 457 x^{12} + 2720 x^{11} + 64336 x^{10} - 240364 x^{9} - 4549640 x^{8} + 4488436 x^{7} + 160506141 x^{6} + 181130646 x^{5} - 2435588247 x^{4} - 6624859400 x^{3} + 8042614275 x^{2} + 46379325210 x + 44040163721 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(438334755667873937053743125776163667968=2^{27}\cdot 7^{12}\cdot 223^{3}\cdot 463^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $575.61$
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:  $2, 7, 223, 463$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{1130506560180518433467823866040754421658170980744942} a^{13} - \frac{267013045763936692427710588130346878928783363528857}{1130506560180518433467823866040754421658170980744942} a^{12} - \frac{106368016398975923302255183823043846527631106761860}{565253280090259216733911933020377210829085490372471} a^{11} + \frac{72827433849894491996625765297173991986840481130023}{565253280090259216733911933020377210829085490372471} a^{10} + \frac{148399123563832697573609409395911022674194486991098}{565253280090259216733911933020377210829085490372471} a^{9} - \frac{23703013036062470517851607532076204449310850288938}{565253280090259216733911933020377210829085490372471} a^{8} - \frac{235363415384441711273967540319530737893902282023531}{565253280090259216733911933020377210829085490372471} a^{7} - \frac{183454589095651239268020241947039772055184643782774}{565253280090259216733911933020377210829085490372471} a^{6} + \frac{260171085530651170449931784429934725382886616848299}{1130506560180518433467823866040754421658170980744942} a^{5} + \frac{1849140973605176171642940364394494572610977357041}{1130506560180518433467823866040754421658170980744942} a^{4} + \frac{145351294876209566228347458423719684741386250901634}{565253280090259216733911933020377210829085490372471} a^{3} + \frac{82149684592629779754628149005562010364844872806666}{565253280090259216733911933020377210829085490372471} a^{2} + \frac{386335965214884325449165028027417041688671731467321}{1130506560180518433467823866040754421658170980744942} a + \frac{477406127518679789201385478973835954021110207194485}{1130506560180518433467823866040754421658170980744942}$

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:  $13$
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:  \( 2024509344240000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T45:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3528
The 35 conjugacy class representatives for [F_42(7)^2]2=F_42(7)wr2
Character table for [F_42(7)^2]2=F_42(7)wr2 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \)

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

Sibling fields

Degree 28 siblings: data not computed
Degree 42 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.12.24.79$x^{12} - 4 x^{11} - 10 x^{10} + 16 x^{9} - 6 x^{8} + 16 x^{7} + 4 x^{6} - 8 x^{5} + 16 x^{4} + 16 x^{3} + 16 x^{2} + 8$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
7.7.7.4$x^{7} + 14 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
223Data not computed
463Data not computed