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Properties

Label 14.14.8016116181...5625.1
Degree $14$
Signature $[14, 0]$
Discriminant $5^{7}\cdot 29^{13}$
Root discriminant $50.98$
Ramified primes $5, 29$
Class number $4$ (GRH)
Class group $[4]$ (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![289, -4386, -1451, 25257, -18628, -16163, 15261, 4383, -4602, -606, 646, 41, -42, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 - 42*x^12 + 41*x^11 + 646*x^10 - 606*x^9 - 4602*x^8 + 4383*x^7 + 15261*x^6 - 16163*x^5 - 18628*x^4 + 25257*x^3 - 1451*x^2 - 4386*x + 289)
 
gp: K = bnfinit(x^14 - x^13 - 42*x^12 + 41*x^11 + 646*x^10 - 606*x^9 - 4602*x^8 + 4383*x^7 + 15261*x^6 - 16163*x^5 - 18628*x^4 + 25257*x^3 - 1451*x^2 - 4386*x + 289, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} - 42 x^{12} + 41 x^{11} + 646 x^{10} - 606 x^{9} - 4602 x^{8} + 4383 x^{7} + 15261 x^{6} - 16163 x^{5} - 18628 x^{4} + 25257 x^{3} - 1451 x^{2} - 4386 x + 289 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(801611618199890796015625=5^{7}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.98$
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:  $5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(145=5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{145}(64,·)$, $\chi_{145}(1,·)$, $\chi_{145}(34,·)$, $\chi_{145}(4,·)$, $\chi_{145}(16,·)$, $\chi_{145}(129,·)$, $\chi_{145}(136,·)$, $\chi_{145}(9,·)$, $\chi_{145}(109,·)$, $\chi_{145}(141,·)$, $\chi_{145}(144,·)$, $\chi_{145}(81,·)$, $\chi_{145}(36,·)$, $\chi_{145}(111,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{17} a^{9} + \frac{5}{17} a^{8} + \frac{6}{17} a^{7} + \frac{1}{17} a^{6} + \frac{8}{17} a^{5} - \frac{7}{17} a^{4} + \frac{1}{17} a^{3} + \frac{4}{17} a^{2} + \frac{4}{17} a$, $\frac{1}{17} a^{10} - \frac{2}{17} a^{8} + \frac{5}{17} a^{7} + \frac{3}{17} a^{6} + \frac{4}{17} a^{5} + \frac{2}{17} a^{4} - \frac{1}{17} a^{3} + \frac{1}{17} a^{2} - \frac{3}{17} a$, $\frac{1}{17} a^{11} - \frac{2}{17} a^{8} - \frac{2}{17} a^{7} + \frac{6}{17} a^{6} + \frac{1}{17} a^{5} + \frac{2}{17} a^{4} + \frac{3}{17} a^{3} + \frac{5}{17} a^{2} + \frac{8}{17} a$, $\frac{1}{1003} a^{12} - \frac{15}{1003} a^{11} + \frac{23}{1003} a^{10} + \frac{6}{1003} a^{9} + \frac{7}{17} a^{8} + \frac{471}{1003} a^{7} + \frac{73}{1003} a^{6} + \frac{143}{1003} a^{5} - \frac{394}{1003} a^{4} + \frac{183}{1003} a^{3} + \frac{481}{1003} a^{2} - \frac{106}{1003} a + \frac{5}{59}$, $\frac{1}{4170455507689} a^{13} + \frac{16600817}{70685686571} a^{12} + \frac{87876081170}{4170455507689} a^{11} + \frac{120359817215}{4170455507689} a^{10} + \frac{4177300980}{4170455507689} a^{9} - \frac{73588596474}{4170455507689} a^{8} + \frac{102819793823}{245320912217} a^{7} + \frac{1146891971111}{4170455507689} a^{6} - \frac{1718148218415}{4170455507689} a^{5} - \frac{836443952500}{4170455507689} a^{4} + \frac{836596380629}{4170455507689} a^{3} + \frac{1348710426515}{4170455507689} a^{2} - \frac{1970416894120}{4170455507689} a + \frac{40183640366}{245320912217}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $13$
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:  \( 9017290.60896 \) (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

\(\Q(\sqrt{145}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$