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Properties

Label 14.0.95706952802...9328.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{12}\cdot 3^{15}\cdot 7^{18}$
Root discriminant $71.74$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T25

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![73872, -341712, 702660, -837144, 646716, -342552, 125097, -33840, 10570, -3528, 567, -112, 42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 42*x^12 - 112*x^11 + 567*x^10 - 3528*x^9 + 10570*x^8 - 33840*x^7 + 125097*x^6 - 342552*x^5 + 646716*x^4 - 837144*x^3 + 702660*x^2 - 341712*x + 73872)
 
gp: K = bnfinit(x^14 + 42*x^12 - 112*x^11 + 567*x^10 - 3528*x^9 + 10570*x^8 - 33840*x^7 + 125097*x^6 - 342552*x^5 + 646716*x^4 - 837144*x^3 + 702660*x^2 - 341712*x + 73872, 1)
 

Normalized defining polynomial

\( x^{14} + 42 x^{12} - 112 x^{11} + 567 x^{10} - 3528 x^{9} + 10570 x^{8} - 33840 x^{7} + 125097 x^{6} - 342552 x^{5} + 646716 x^{4} - 837144 x^{3} + 702660 x^{2} - 341712 x + 73872 \)

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:  \(-95706952802109141111779328=-\,2^{12}\cdot 3^{15}\cdot 7^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.74$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{28} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{2}{7}$, $\frac{1}{28} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{2}{7} a$, $\frac{1}{56} a^{9} - \frac{1}{56} a^{8} + \frac{1}{8} a^{6} + \frac{1}{8} a^{4} + \frac{3}{8} a^{3} + \frac{3}{28} a^{2} - \frac{3}{28} a - \frac{1}{2}$, $\frac{1}{56} a^{10} - \frac{1}{56} a^{8} - \frac{1}{56} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{27}{56} a^{3} + \frac{11}{28} a - \frac{5}{14}$, $\frac{1}{56} a^{11} - \frac{1}{56} a^{7} + \frac{5}{14} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{5}{14}$, $\frac{1}{336} a^{12} + \frac{1}{168} a^{9} - \frac{1}{112} a^{8} + \frac{5}{24} a^{6} - \frac{3}{28} a^{5} + \frac{7}{16} a^{4} - \frac{1}{8} a^{3} - \frac{5}{56} a^{2} - \frac{3}{7} a - \frac{1}{2}$, $\frac{1}{43847636366125315584} a^{13} + \frac{444155934080723}{2435979798118073088} a^{12} + \frac{52114212260087821}{7307939394354219264} a^{11} - \frac{65403855934071967}{10961909091531328896} a^{10} - \frac{17330675450250217}{4871959596236146176} a^{9} - \frac{9363717722097511}{2435979798118073088} a^{8} - \frac{41132239833619}{3131974026151808256} a^{7} + \frac{50113356316409255}{405996633019678848} a^{6} - \frac{1954715697062630197}{14615878788708438528} a^{5} - \frac{1395861894971918987}{7307939394354219264} a^{4} - \frac{104852412336040553}{913492424294277408} a^{3} + \frac{88772041199418289}{608994949529518272} a^{2} + \frac{1463207729815260703}{3653969697177109632} a + \frac{533538100489747}{1526303131652928}$

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:  \( -\frac{13043977103}{36377785511424} a^{13} - \frac{825350605}{2020988083968} a^{12} - \frac{13304665829}{866137750272} a^{11} + \frac{210282497633}{9094446377856} a^{10} - \frac{97631475679}{577425166848} a^{9} + \frac{2173083511289}{2020988083968} a^{8} - \frac{6438780684547}{2598413250816} a^{7} + \frac{3021239495863}{336831347328} a^{6} - \frac{412360315127077}{12125928503808} a^{5} + \frac{70043009936035}{866137750272} a^{4} - \frac{97941467952833}{757870531488} a^{3} + \frac{9495787540975}{72178145856} a^{2} - \frac{221766616259633}{3031482125952} a + \frac{422187193753}{24059381952} \) (order $6$)
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:  \( 596110111.893 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T25:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 588
The 19 conjugacy class representatives for [7^2:6_3]2
Character table for [7^2:6_3]2

Intermediate fields

\(\Q(\sqrt{-3}) \)

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

Sibling fields

Degree 21 siblings: data not computed
Degree 28 sibling: 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 R ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$7$7.7.7.1$x^{7} + 42 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
7.7.11.2$x^{7} + 28 x^{5} + 7$$7$$1$$11$$F_7$$[11/6]_{6}$