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Properties

Label 14.0.34935649744...3483.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,23^{7}\cdot 29^{13}$
Root discriminant $109.35$
Ramified primes $23, 29$
Class number $68128$ (GRH)
Class group $[2, 2, 2, 8516]$ (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![41183711, -47391691, 48954232, -18171460, 9051006, -4073118, 1855050, -261750, 199413, -8929, 8766, -162, 161, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 161*x^12 - 162*x^11 + 8766*x^10 - 8929*x^9 + 199413*x^8 - 261750*x^7 + 1855050*x^6 - 4073118*x^5 + 9051006*x^4 - 18171460*x^3 + 48954232*x^2 - 47391691*x + 41183711)
 
gp: K = bnfinit(x^14 - x^13 + 161*x^12 - 162*x^11 + 8766*x^10 - 8929*x^9 + 199413*x^8 - 261750*x^7 + 1855050*x^6 - 4073118*x^5 + 9051006*x^4 - 18171460*x^3 + 48954232*x^2 - 47391691*x + 41183711, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 161 x^{12} - 162 x^{11} + 8766 x^{10} - 8929 x^{9} + 199413 x^{8} - 261750 x^{7} + 1855050 x^{6} - 4073118 x^{5} + 9051006 x^{4} - 18171460 x^{3} + 48954232 x^{2} - 47391691 x + 41183711 \)

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:  \(-34935649744100307390657103483=-\,23^{7}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $109.35$
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:  $23, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(667=23\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{667}(576,·)$, $\chi_{667}(1,·)$, $\chi_{667}(643,·)$, $\chi_{667}(484,·)$, $\chi_{667}(645,·)$, $\chi_{667}(390,·)$, $\chi_{667}(139,·)$, $\chi_{667}(528,·)$, $\chi_{667}(277,·)$, $\chi_{667}(22,·)$, $\chi_{667}(183,·)$, $\chi_{667}(24,·)$, $\chi_{667}(666,·)$, $\chi_{667}(91,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{59} a^{11} - \frac{8}{59} a^{10} + \frac{16}{59} a^{9} - \frac{23}{59} a^{8} - \frac{12}{59} a^{7} + \frac{26}{59} a^{6} - \frac{19}{59} a^{5} + \frac{14}{59} a^{4} - \frac{24}{59} a^{3} + \frac{15}{59} a^{2} - \frac{8}{59} a$, $\frac{1}{1003} a^{12} + \frac{4}{1003} a^{11} - \frac{316}{1003} a^{10} - \frac{126}{1003} a^{9} - \frac{52}{1003} a^{8} - \frac{4}{17} a^{7} - \frac{14}{59} a^{6} - \frac{332}{1003} a^{5} + \frac{203}{1003} a^{4} + \frac{317}{1003} a^{3} + \frac{231}{1003} a^{2} - \frac{273}{1003} a - \frac{2}{17}$, $\frac{1}{5808334138971223099257033557657992562788161967} a^{13} + \frac{158275880788793608706608078423857678100916}{341666714057130770544531385744587797811068351} a^{12} + \frac{4385833104191863632849419576631863435926304}{5808334138971223099257033557657992562788161967} a^{11} + \frac{2529806905694332561014371975893120204047636387}{5808334138971223099257033557657992562788161967} a^{10} + \frac{2815411330416041398166753407397190180912551974}{5808334138971223099257033557657992562788161967} a^{9} + \frac{1435498251022983877555183913795477603828275172}{5808334138971223099257033557657992562788161967} a^{8} + \frac{1714194754819213292359836800901952384368064228}{5808334138971223099257033557657992562788161967} a^{7} - \frac{2559459035355987746146118932848091686783661057}{5808334138971223099257033557657992562788161967} a^{6} - \frac{1053535808744336160161381895821868422914109037}{5808334138971223099257033557657992562788161967} a^{5} - \frac{30802454833878954898583881543894816093152703}{98446341338495306767068365384033772250646813} a^{4} - \frac{154456940151276272483075580778104605947488708}{341666714057130770544531385744587797811068351} a^{3} + \frac{1268334839589452811531277485211369504843426089}{5808334138971223099257033557657992562788161967} a^{2} - \frac{941062387284609131874283748615749851646554849}{5808334138971223099257033557657992562788161967} a + \frac{41611446743511598506695024961276629159336730}{98446341338495306767068365384033772250646813}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{8516}$, which has order $68128$ (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

\(\Q(\sqrt{-667}) \), 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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
$23$23.14.7.2$x^{14} - 148035889 x^{2} + 27238603576$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$