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Properties

Label 14.0.56430204326...7168.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 829^{7}$
Root discriminant $81.44$
Ramified primes $2, 829$
Class number $1014$ (GRH)
Class group $[13, 78]$ (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![106112, 0, 692068, 0, 26604, 0, 46193, 0, 4936, 0, 122, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^12 + 122*x^10 + 4936*x^8 + 46193*x^6 + 26604*x^4 + 692068*x^2 + 106112)
 
gp: K = bnfinit(x^14 - 4*x^12 + 122*x^10 + 4936*x^8 + 46193*x^6 + 26604*x^4 + 692068*x^2 + 106112, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{12} + 122 x^{10} + 4936 x^{8} + 46193 x^{6} + 26604 x^{4} + 692068 x^{2} + 106112 \)

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:  \(-564302043262742500808327168=-\,2^{21}\cdot 829^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.44$
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:  $2, 829$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2320} a^{10} - \frac{137}{2320} a^{8} + \frac{269}{2320} a^{6} + \frac{17}{2320} a^{4} - \frac{1}{2} a^{3} + \frac{333}{1160} a^{2} - \frac{1}{2} a - \frac{22}{145}$, $\frac{1}{4640} a^{11} - \frac{137}{4640} a^{9} + \frac{269}{4640} a^{7} - \frac{1143}{4640} a^{5} - \frac{1}{4} a^{4} + \frac{913}{2320} a^{3} - \frac{1}{4} a^{2} - \frac{11}{145} a$, $\frac{1}{283416863959840} a^{12} + \frac{23964157803}{283416863959840} a^{10} - \frac{11616494726151}{283416863959840} a^{8} - \frac{5870997257963}{283416863959840} a^{6} - \frac{636059859269}{10900648613840} a^{4} - \frac{1}{2} a^{3} + \frac{12418948142241}{35427107994980} a^{2} - \frac{1}{2} a - \frac{251968934879}{1771355399749}$, $\frac{1}{566833727919680} a^{13} - \frac{1}{566833727919680} a^{12} + \frac{23964157803}{566833727919680} a^{11} + \frac{98198283559}{566833727919680} a^{10} - \frac{11616494726151}{566833727919680} a^{9} - \frac{5119759740443}{566833727919680} a^{8} - \frac{5870997257963}{566833727919680} a^{7} + \frac{38732693984341}{566833727919680} a^{6} + \frac{4814264447651}{21801297227680} a^{5} - \frac{2367194502611}{10900648613840} a^{4} - \frac{5294605855249}{70854215989960} a^{3} + \frac{30929258197271}{141708431979920} a^{2} - \frac{251968934879}{3542710799498} a + \frac{4386417409079}{8856776998745}$

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

Class group and class number

$C_{13}\times C_{78}$, which has order $1014$ (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:  \( 4905723.82269 \) (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

\(\Q(\sqrt{-1658}) \), 7.1.291698067968.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.291698067968.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
829Data not computed