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Properties

Label 14.0.35718784670...1339.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,19^{7}\cdot 43^{12}$
Root discriminant $109.52$
Ramified primes $19, 43$
Class number $24059$ (GRH)
Class group $[7, 3437]$ (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![3303433, 1206310, 1450369, 197546, 293844, 25532, 43763, -1563, 3663, -307, 380, -45, 8, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 8*x^12 - 45*x^11 + 380*x^10 - 307*x^9 + 3663*x^8 - 1563*x^7 + 43763*x^6 + 25532*x^5 + 293844*x^4 + 197546*x^3 + 1450369*x^2 + 1206310*x + 3303433)
 
gp: K = bnfinit(x^14 - 5*x^13 + 8*x^12 - 45*x^11 + 380*x^10 - 307*x^9 + 3663*x^8 - 1563*x^7 + 43763*x^6 + 25532*x^5 + 293844*x^4 + 197546*x^3 + 1450369*x^2 + 1206310*x + 3303433, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + 8 x^{12} - 45 x^{11} + 380 x^{10} - 307 x^{9} + 3663 x^{8} - 1563 x^{7} + 43763 x^{6} + 25532 x^{5} + 293844 x^{4} + 197546 x^{3} + 1450369 x^{2} + 1206310 x + 3303433 \)

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:  \(-35718784670547055607182231339=-\,19^{7}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $109.52$
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:  $19, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(817=19\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{817}(704,·)$, $\chi_{817}(1,·)$, $\chi_{817}(514,·)$, $\chi_{817}(742,·)$, $\chi_{817}(360,·)$, $\chi_{817}(778,·)$, $\chi_{817}(322,·)$, $\chi_{817}(398,·)$, $\chi_{817}(305,·)$, $\chi_{817}(723,·)$, $\chi_{817}(666,·)$, $\chi_{817}(379,·)$, $\chi_{817}(170,·)$, $\chi_{817}(474,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{5}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{6}$, $\frac{1}{91258438864113461544275384111} a^{13} - \frac{3239824807467908528054067696}{91258438864113461544275384111} a^{12} - \frac{3801074143209921507666291644}{91258438864113461544275384111} a^{11} + \frac{6467129828067949921616135877}{91258438864113461544275384111} a^{10} + \frac{5755295817013412811924001349}{91258438864113461544275384111} a^{9} + \frac{76327706082615873294765120}{91258438864113461544275384111} a^{8} - \frac{3486033383383871581711612464}{91258438864113461544275384111} a^{7} + \frac{12301411775411224698562876689}{91258438864113461544275384111} a^{6} - \frac{12937596446897920094177977084}{91258438864113461544275384111} a^{5} + \frac{13143845848581650711436943896}{91258438864113461544275384111} a^{4} + \frac{17415704700958975339831577118}{91258438864113461544275384111} a^{3} - \frac{19287527689672873966735561450}{91258438864113461544275384111} a^{2} + \frac{4893513548376325864907752485}{13036919837730494506325054873} a + \frac{57526220793786799253200810}{266059588525112132782143977}$

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

Class group and class number

$C_{7}\times C_{3437}$, which has order $24059$ (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:  \( 35991.64185055774 \) (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

\(\Q(\sqrt{-19}) \), 7.7.6321363049.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$19$19.14.7.2$x^{14} - 376367048 x^{2} + 3575486956$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$43$43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$