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Properties

Label 14.14.2764177993...8125.1
Degree $14$
Signature $[14, 0]$
Discriminant $5^{7}\cdot 29^{12}$
Root discriminant $40.08$
Ramified primes $5, 29$
Class number $1$ (GRH)
Class group Trivial (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![41, -168, -781, 1404, 2916, -4170, -1853, 3253, 223, -1019, 90, 131, -22, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 22*x^12 + 131*x^11 + 90*x^10 - 1019*x^9 + 223*x^8 + 3253*x^7 - 1853*x^6 - 4170*x^5 + 2916*x^4 + 1404*x^3 - 781*x^2 - 168*x + 41)
 
gp: K = bnfinit(x^14 - 5*x^13 - 22*x^12 + 131*x^11 + 90*x^10 - 1019*x^9 + 223*x^8 + 3253*x^7 - 1853*x^6 - 4170*x^5 + 2916*x^4 + 1404*x^3 - 781*x^2 - 168*x + 41, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - 22 x^{12} + 131 x^{11} + 90 x^{10} - 1019 x^{9} + 223 x^{8} + 3253 x^{7} - 1853 x^{6} - 4170 x^{5} + 2916 x^{4} + 1404 x^{3} - 781 x^{2} - 168 x + 41 \)

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:  \(27641779937927268828125=5^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.08$
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$$\chi_{145}(1,·)$, $\chi_{145}(36,·)$, $\chi_{145}(81,·)$, $\chi_{145}(136,·)$, $\chi_{145}(74,·)$, $\chi_{145}(139,·)$, $\chi_{145}(141,·)$, $\chi_{145}(111,·)$, $\chi_{145}(16,·)$, $\chi_{145}(49,·)$, $\chi_{145}(54,·)$, $\chi_{145}(24,·)$, $\chi_{145}(59,·)$, $\chi_{145}(94,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{41123} a^{12} + \frac{1384}{41123} a^{11} + \frac{11634}{41123} a^{10} + \frac{17576}{41123} a^{9} - \frac{4877}{41123} a^{8} + \frac{5492}{41123} a^{7} + \frac{7127}{41123} a^{6} + \frac{2667}{41123} a^{5} + \frac{17058}{41123} a^{4} - \frac{7782}{41123} a^{3} + \frac{9032}{41123} a^{2} + \frac{240}{697} a - \frac{401}{1003}$, $\frac{1}{11639412797} a^{13} + \frac{90482}{11639412797} a^{12} + \frac{949854443}{11639412797} a^{11} - \frac{2875036052}{11639412797} a^{10} - \frac{204989855}{684671341} a^{9} + \frac{2638226229}{11639412797} a^{8} - \frac{3790954112}{11639412797} a^{7} + \frac{4458168300}{11639412797} a^{6} - \frac{561501835}{11639412797} a^{5} + \frac{4552893890}{11639412797} a^{4} + \frac{1130206708}{11639412797} a^{3} + \frac{4942502433}{11639412797} a^{2} - \frac{3552699494}{11639412797} a - \frac{83551338}{283888117}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 4233583.6638 \) (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

\(\Q(\sqrt{5}) \), 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ R ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$