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Properties

Label 14.0.53588606915...5607.2
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 43^{12}$
Root discriminant $811.37$
Ramified primes $7, 43$
Class number $1209467$ (GRH)
Class group $[7, 172781]$ (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16680217258567, -164577661339, 1628162950638, 49362000457, 7724343571, 1542753828, -935002320, -95951318, 16307879, 176386, 25585, -7224, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7224*x^11 + 25585*x^10 + 176386*x^9 + 16307879*x^8 - 95951318*x^7 - 935002320*x^6 + 1542753828*x^5 + 7724343571*x^4 + 49362000457*x^3 + 1628162950638*x^2 - 164577661339*x + 16680217258567)
 
gp: K = bnfinit(x^14 - 7224*x^11 + 25585*x^10 + 176386*x^9 + 16307879*x^8 - 95951318*x^7 - 935002320*x^6 + 1542753828*x^5 + 7724343571*x^4 + 49362000457*x^3 + 1628162950638*x^2 - 164577661339*x + 16680217258567, 1)
 

Normalized defining polynomial

\( x^{14} - 7224 x^{11} + 25585 x^{10} + 176386 x^{9} + 16307879 x^{8} - 95951318 x^{7} - 935002320 x^{6} + 1542753828 x^{5} + 7724343571 x^{4} + 49362000457 x^{3} + 1628162950638 x^{2} - 164577661339 x + 16680217258567 \)

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:  \(-53588606915566584613059849086223224055607=-\,7^{25}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $811.37$
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:  $7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2107=7^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2107}(1,·)$, $\chi_{2107}(1091,·)$, $\chi_{2107}(1798,·)$, $\chi_{2107}(778,·)$, $\chi_{2107}(1420,·)$, $\chi_{2107}(1933,·)$, $\chi_{2107}(1583,·)$, $\chi_{2107}(692,·)$, $\chi_{2107}(1784,·)$, $\chi_{2107}(666,·)$, $\chi_{2107}(1903,·)$, $\chi_{2107}(1546,·)$, $\chi_{2107}(1086,·)$, $\chi_{2107}(575,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{43} a^{7}$, $\frac{1}{43} a^{8}$, $\frac{1}{43} a^{9}$, $\frac{1}{42527} a^{10} - \frac{10}{989} a^{9} + \frac{3}{989} a^{8} - \frac{4}{989} a^{7} - \frac{136}{989} a^{6} - \frac{413}{989} a^{5} + \frac{380}{989} a^{4} - \frac{285}{989} a^{3} - \frac{2}{23} a^{2} - \frac{3}{23} a$, $\frac{1}{42527} a^{11} + \frac{4}{989} a^{9} - \frac{2}{989} a^{8} + \frac{7}{989} a^{7} + \frac{447}{989} a^{6} - \frac{179}{989} a^{5} - \frac{70}{989} a^{4} + \frac{11}{23} a^{2} - \frac{2}{23} a$, $\frac{1}{1828661} a^{12} - \frac{168}{42527} a^{9} - \frac{394}{42527} a^{8} + \frac{146}{42527} a^{7} - \frac{3490}{42527} a^{6} - \frac{20022}{42527} a^{5} - \frac{7}{23} a^{4} - \frac{8}{23} a^{3} + \frac{1}{23} a^{2} - \frac{11}{23} a$, $\frac{1}{114969514977201829529130453418861331144279640157034257016243} a^{13} + \frac{231522127948783737642192339315147767059374447389904}{4998674564226166501266541452993970919316506093784098131141} a^{12} - \frac{210888548447487536505083721586473606207752013553434}{116248245679678290727128870999859788821314095204281351887} a^{11} + \frac{297033220754843938425268303698713552426475682837653}{62179294200758155505208465883645933555586609062755141707} a^{10} - \frac{28894368985814657188805421644001057413482977806699661600}{2673709650632600686723964032996775142890224189698471093401} a^{9} + \frac{5809682819791558182928101785892209296739306621514801705}{2673709650632600686723964032996775142890224189698471093401} a^{8} + \frac{4050599085331539241339920371435539550120492258891805937}{2673709650632600686723964032996775142890224189698471093401} a^{7} + \frac{15697312766723416546501446419740839581815745052572880946}{2673709650632600686723964032996775142890224189698471093401} a^{6} - \frac{719183389511256603709389639836135411688189411095352927250}{2673709650632600686723964032996775142890224189698471093401} a^{5} - \frac{16385358927454123343230602903231734780784953657250976610}{62179294200758155505208465883645933555586609062755141707} a^{4} + \frac{23198693446777514611949017836630151122842562079236581220}{62179294200758155505208465883645933555586609062755141707} a^{3} - \frac{510003929489493714406897392968580294194401036732282384}{1446030097692050128028103857759207757106665327040817249} a^{2} + \frac{11590636739268621423810401414578004433009397864343555}{62870873812697831653395819902574250308985449001774663} a - \frac{1114395958041538721855037474401560031422995698222993}{2733516252725992680582426952285836969955889087033681}$

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

Class group and class number

$C_{7}\times C_{172781}$, which has order $1209467$ (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:  \( 301031344.50909585 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-7}) \), 7.7.87495801462998035849.5

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ R ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$7$7.14.25.59$x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$$14$$1$$25$$C_{14}$$[2]_{2}$
$43$43.7.6.6$x^{7} + 2539107$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.6$x^{7} + 2539107$$7$$1$$6$$C_7$$[\ ]_{7}$