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Properties

Label 14.0.33289688362...3739.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,379^{13}$
Root discriminant $248.00$
Ramified prime $379$
Class number $4497$ (GRH)
Class group $[4497]$ (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![148699319, -10003960, -11883459, 17073682, 2562883, -3184843, -186895, 300557, -8271, -19828, 840, 754, 14, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 14*x^12 + 754*x^11 + 840*x^10 - 19828*x^9 - 8271*x^8 + 300557*x^7 - 186895*x^6 - 3184843*x^5 + 2562883*x^4 + 17073682*x^3 - 11883459*x^2 - 10003960*x + 148699319)
 
gp: K = bnfinit(x^14 - x^13 + 14*x^12 + 754*x^11 + 840*x^10 - 19828*x^9 - 8271*x^8 + 300557*x^7 - 186895*x^6 - 3184843*x^5 + 2562883*x^4 + 17073682*x^3 - 11883459*x^2 - 10003960*x + 148699319, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 14 x^{12} + 754 x^{11} + 840 x^{10} - 19828 x^{9} - 8271 x^{8} + 300557 x^{7} - 186895 x^{6} - 3184843 x^{5} + 2562883 x^{4} + 17073682 x^{3} - 11883459 x^{2} - 10003960 x + 148699319 \)

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:  \(-3328968836295431565149961657333739=-\,379^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $248.00$
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:  $379$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(379\)
Dirichlet character group:    $\lbrace$$\chi_{379}(1,·)$, $\chi_{379}(195,·)$, $\chi_{379}(260,·)$, $\chi_{379}(293,·)$, $\chi_{379}(138,·)$, $\chi_{379}(125,·)$, $\chi_{379}(241,·)$, $\chi_{379}(94,·)$, $\chi_{379}(86,·)$, $\chi_{379}(119,·)$, $\chi_{379}(184,·)$, $\chi_{379}(378,·)$, $\chi_{379}(285,·)$, $\chi_{379}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5539985449407491867495356759667961063719262944971} a^{13} - \frac{2181491925514533352020617621687893304219474131717}{5539985449407491867495356759667961063719262944971} a^{12} - \frac{572370004079953758602310915795848280120667907927}{5539985449407491867495356759667961063719262944971} a^{11} + \frac{1138146363352398862872606904482074207200840546609}{5539985449407491867495356759667961063719262944971} a^{10} - \frac{2256597505110529233098043543053725218569470878465}{5539985449407491867495356759667961063719262944971} a^{9} - \frac{36581693064367291289375278519162087412629599921}{90819433596844128975333717371605919077364966311} a^{8} - \frac{1446819822232959032303270414534959496792576757387}{5539985449407491867495356759667961063719262944971} a^{7} + \frac{1534048819665959610684139356101311113864594104269}{5539985449407491867495356759667961063719262944971} a^{6} - \frac{2583904749397210853347367543593486751882261383293}{5539985449407491867495356759667961063719262944971} a^{5} + \frac{286795985732914791878519094086010488620198747847}{5539985449407491867495356759667961063719262944971} a^{4} - \frac{7233134088671555213887976212128427586319599664}{27839122861344180238670134470693271676981220829} a^{3} - \frac{2420452941084115914639191180701138579250565303355}{5539985449407491867495356759667961063719262944971} a^{2} + \frac{41119516369265257954250849338722333064458480308}{5539985449407491867495356759667961063719262944971} a + \frac{2097572508975540493223282964926621919741596384529}{5539985449407491867495356759667961063719262944971}$

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

Class group and class number

$C_{4497}$, which has order $4497$ (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:  \( 17223236.102860697 \) (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

\(\Q(\sqrt{-379}) \), 7.7.2963706958323721.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
379Data not computed