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Properties

Label 15.15.3659245464...5529.1
Degree $15$
Signature $[15, 0]$
Discriminant $13^{10}\cdot 61^{12}$
Root discriminant $148.22$
Ramified primes $13, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{15}$ (as 15T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18863, -121589, 5239, 594218, -14194, -804863, -122146, 276726, 37495, -41558, -3738, 2960, 144, -93, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 93*x^13 + 144*x^12 + 2960*x^11 - 3738*x^10 - 41558*x^9 + 37495*x^8 + 276726*x^7 - 122146*x^6 - 804863*x^5 - 14194*x^4 + 594218*x^3 + 5239*x^2 - 121589*x + 18863)
 
gp: K = bnfinit(x^15 - 2*x^14 - 93*x^13 + 144*x^12 + 2960*x^11 - 3738*x^10 - 41558*x^9 + 37495*x^8 + 276726*x^7 - 122146*x^6 - 804863*x^5 - 14194*x^4 + 594218*x^3 + 5239*x^2 - 121589*x + 18863, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 93 x^{13} + 144 x^{12} + 2960 x^{11} - 3738 x^{10} - 41558 x^{9} + 37495 x^{8} + 276726 x^{7} - 122146 x^{6} - 804863 x^{5} - 14194 x^{4} + 594218 x^{3} + 5239 x^{2} - 121589 x + 18863 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(365924546437605291907270802025529=13^{10}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $148.22$
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:  $13, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(793=13\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{793}(672,·)$, $\chi_{793}(1,·)$, $\chi_{793}(386,·)$, $\chi_{793}(131,·)$, $\chi_{793}(81,·)$, $\chi_{793}(705,·)$, $\chi_{793}(9,·)$, $\chi_{793}(302,·)$, $\chi_{793}(367,·)$, $\chi_{793}(497,·)$, $\chi_{793}(339,·)$, $\chi_{793}(729,·)$, $\chi_{793}(217,·)$, $\chi_{793}(508,·)$, $\chi_{793}(607,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{29} a^{12} + \frac{3}{29} a^{11} + \frac{5}{29} a^{10} + \frac{4}{29} a^{8} - \frac{8}{29} a^{7} + \frac{1}{29} a^{6} - \frac{13}{29} a^{5} + \frac{6}{29} a^{4} + \frac{14}{29} a^{3} + \frac{2}{29} a^{2} + \frac{14}{29} a - \frac{11}{29}$, $\frac{1}{29} a^{13} - \frac{4}{29} a^{11} + \frac{14}{29} a^{10} + \frac{4}{29} a^{9} + \frac{9}{29} a^{8} - \frac{4}{29} a^{7} + \frac{13}{29} a^{6} - \frac{13}{29} a^{5} - \frac{4}{29} a^{4} - \frac{11}{29} a^{3} + \frac{8}{29} a^{2} + \frac{5}{29} a + \frac{4}{29}$, $\frac{1}{5662037402009775776613726818869} a^{14} - \frac{33288314877623599836300901583}{5662037402009775776613726818869} a^{13} - \frac{1864853268665625340703073088}{195242669034819854365990579961} a^{12} - \frac{542960388338890476193353544060}{5662037402009775776613726818869} a^{11} - \frac{2136893500160437884946461903255}{5662037402009775776613726818869} a^{10} + \frac{2600116591390834205641518262385}{5662037402009775776613726818869} a^{9} + \frac{1292956835081908188656964678278}{5662037402009775776613726818869} a^{8} + \frac{1954570635177293091704225691524}{5662037402009775776613726818869} a^{7} - \frac{1722701202096359727360465135360}{5662037402009775776613726818869} a^{6} + \frac{64989745915757681514444671172}{5662037402009775776613726818869} a^{5} + \frac{1299892900195203571508813793524}{5662037402009775776613726818869} a^{4} - \frac{561505825083476190541256722069}{5662037402009775776613726818869} a^{3} + \frac{1149824247358465459697550769449}{5662037402009775776613726818869} a^{2} - \frac{2529748324466721385513734754971}{5662037402009775776613726818869} a - \frac{1398755347265996755267303981336}{5662037402009775776613726818869}$

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:  $14$
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:  \( 322714450041.6809 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{15}$ (as 15T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 15
The 15 conjugacy class representatives for $C_{15}$
Character table for $C_{15}$

Intermediate fields

3.3.169.1, 5.5.13845841.1

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 $15$ $15$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ R $15$ $15$ $15$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ $15$ $15$ $15$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
$61$61.15.12.1$x^{15} + 3050 x^{10} + 1856779 x^{5} + 22698100000$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$