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Properties

Label 14.0.68948491826...3011.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,11^{7}\cdot 29^{12}$
Root discriminant $59.45$
Ramified primes $11, 29$
Class number $344$ (GRH)
Class group $[2, 2, 86]$ (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![130153, -68096, 108523, -47572, 45292, -18010, 9895, -3203, 1275, -315, 54, 11, 6, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 6*x^12 + 11*x^11 + 54*x^10 - 315*x^9 + 1275*x^8 - 3203*x^7 + 9895*x^6 - 18010*x^5 + 45292*x^4 - 47572*x^3 + 108523*x^2 - 68096*x + 130153)
 
gp: K = bnfinit(x^14 - 5*x^13 + 6*x^12 + 11*x^11 + 54*x^10 - 315*x^9 + 1275*x^8 - 3203*x^7 + 9895*x^6 - 18010*x^5 + 45292*x^4 - 47572*x^3 + 108523*x^2 - 68096*x + 130153, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + 6 x^{12} + 11 x^{11} + 54 x^{10} - 315 x^{9} + 1275 x^{8} - 3203 x^{7} + 9895 x^{6} - 18010 x^{5} + 45292 x^{4} - 47572 x^{3} + 108523 x^{2} - 68096 x + 130153 \)

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:  \(-6894849182652903337173011=-\,11^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.45$
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:  $11, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(319=11\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{319}(1,·)$, $\chi_{319}(197,·)$, $\chi_{319}(65,·)$, $\chi_{319}(199,·)$, $\chi_{319}(45,·)$, $\chi_{319}(78,·)$, $\chi_{319}(175,·)$, $\chi_{319}(210,·)$, $\chi_{319}(54,·)$, $\chi_{319}(23,·)$, $\chi_{319}(111,·)$, $\chi_{319}(219,·)$, $\chi_{319}(252,·)$, $\chi_{319}(285,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} + \frac{7}{17} a^{11} - \frac{4}{17} a^{10} + \frac{2}{17} a^{9} - \frac{5}{17} a^{8} - \frac{2}{17} a^{7} + \frac{4}{17} a^{6} + \frac{8}{17} a^{5} - \frac{7}{17} a^{4} + \frac{7}{17} a^{3} - \frac{2}{17} a^{2} - \frac{8}{17} a + \frac{2}{17}$, $\frac{1}{2703706331164251614721780343} a^{13} - \frac{12495713740811643920422868}{2703706331164251614721780343} a^{12} + \frac{21799534981827916974526249}{2703706331164251614721780343} a^{11} + \frac{58046986896704241508151606}{159041548892014800865987079} a^{10} + \frac{1176246536709448985155399508}{2703706331164251614721780343} a^{9} - \frac{857344167073994860503329568}{2703706331164251614721780343} a^{8} + \frac{398203165457311843570039441}{2703706331164251614721780343} a^{7} + \frac{505051682991829025694187193}{2703706331164251614721780343} a^{6} - \frac{1323791263471766707539168114}{2703706331164251614721780343} a^{5} + \frac{1022361781605015273771466466}{2703706331164251614721780343} a^{4} + \frac{790643291372915250313176263}{2703706331164251614721780343} a^{3} - \frac{286195546878460035577917037}{2703706331164251614721780343} a^{2} + \frac{446197926551148805413920929}{2703706331164251614721780343} a - \frac{171455554545250476069548423}{2703706331164251614721780343}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{86}$, which has order $344$ (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:  \( 6020.985100147561 \) (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

\(\Q(\sqrt{-11}) \), 7.7.594823321.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ R ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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
$11$11.14.7.2$x^{14} - 1771561 x^{2} + 77948684$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$