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Properties

Label 14.0.28137207545...5063.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 197^{12}$
Root discriminant $245.04$
Ramified primes $7, 197$
Class number $27496$ (GRH)
Class group $[2, 2, 6874]$ (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![101385859, -15092353, 45137186, 10375639, -3383529, 5538516, 2594626, -314915, -234152, 200, 8850, 341, -145, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 145*x^12 + 341*x^11 + 8850*x^10 + 200*x^9 - 234152*x^8 - 314915*x^7 + 2594626*x^6 + 5538516*x^5 - 3383529*x^4 + 10375639*x^3 + 45137186*x^2 - 15092353*x + 101385859)
 
gp: K = bnfinit(x^14 - 5*x^13 - 145*x^12 + 341*x^11 + 8850*x^10 + 200*x^9 - 234152*x^8 - 314915*x^7 + 2594626*x^6 + 5538516*x^5 - 3383529*x^4 + 10375639*x^3 + 45137186*x^2 - 15092353*x + 101385859, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - 145 x^{12} + 341 x^{11} + 8850 x^{10} + 200 x^{9} - 234152 x^{8} - 314915 x^{7} + 2594626 x^{6} + 5538516 x^{5} - 3383529 x^{4} + 10375639 x^{3} + 45137186 x^{2} - 15092353 x + 101385859 \)

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:  \(-2813720754599325443123783814625063=-\,7^{7}\cdot 197^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $245.04$
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:  $7, 197$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1379=7\cdot 197\)
Dirichlet character group:    $\lbrace$$\chi_{1379}(1,·)$, $\chi_{1379}(36,·)$, $\chi_{1379}(902,·)$, $\chi_{1379}(769,·)$, $\chi_{1379}(104,·)$, $\chi_{1379}(1163,·)$, $\chi_{1379}(1021,·)$, $\chi_{1379}(1296,·)$, $\chi_{1379}(498,·)$, $\chi_{1379}(755,·)$, $\chi_{1379}(1373,·)$, $\chi_{1379}(986,·)$, $\chi_{1379}(1149,·)$, $\chi_{1379}(979,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{1477387} a^{12} - \frac{185587}{1477387} a^{11} - \frac{305542}{1477387} a^{10} - \frac{90908}{1477387} a^{9} - \frac{258307}{1477387} a^{8} + \frac{56637}{1477387} a^{7} + \frac{704318}{1477387} a^{6} - \frac{346360}{1477387} a^{5} + \frac{583768}{1477387} a^{4} + \frac{70375}{1477387} a^{3} - \frac{380405}{1477387} a^{2} + \frac{387111}{1477387} a - \frac{724540}{1477387}$, $\frac{1}{4395163787734732340440614162929581343101} a^{13} - \frac{261379508796657549843140270475883}{4395163787734732340440614162929581343101} a^{12} + \frac{1150970480616523361548285784248383571339}{4395163787734732340440614162929581343101} a^{11} - \frac{1889849143103183090994904035386177282679}{4395163787734732340440614162929581343101} a^{10} + \frac{1213057173100697246220696283664650065038}{4395163787734732340440614162929581343101} a^{9} - \frac{331560950835393929937887882277399261273}{4395163787734732340440614162929581343101} a^{8} - \frac{1661506887846466964690840412456315250225}{4395163787734732340440614162929581343101} a^{7} + \frac{2194715789385806079634897352129857199687}{4395163787734732340440614162929581343101} a^{6} - \frac{1594702428355500398021753697081200571593}{4395163787734732340440614162929581343101} a^{5} - \frac{1408006213925067488554893242403838221157}{4395163787734732340440614162929581343101} a^{4} + \frac{459616789480145245462981139091106607448}{4395163787734732340440614162929581343101} a^{3} + \frac{144350099514161259701503388148002222179}{4395163787734732340440614162929581343101} a^{2} + \frac{1878285448059991462608176642342332744837}{4395163787734732340440614162929581343101} a - \frac{1550854145896814334345613352376808643699}{4395163787734732340440614162929581343101}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6874}$, which has order $27496$ (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:  \( 1553055.199048291 \) (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

\(\Q(\sqrt{-7}) \), 7.7.58451728309129.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$197$197.7.6.1$x^{7} - 197$$7$$1$$6$$C_7$$[\ ]_{7}$
197.7.6.1$x^{7} - 197$$7$$1$$6$$C_7$$[\ ]_{7}$