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Properties

Label 15.15.2866139633...5625.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{24}\cdot 37^{10}$
Root discriminant $145.82$
Ramified primes $5, 37$
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![-477401, -1278880, 419510, 2166295, 144735, -1227567, -181935, 314635, 51015, -39850, -5909, 2545, 290, -80, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 80*x^13 + 290*x^12 + 2545*x^11 - 5909*x^10 - 39850*x^9 + 51015*x^8 + 314635*x^7 - 181935*x^6 - 1227567*x^5 + 144735*x^4 + 2166295*x^3 + 419510*x^2 - 1278880*x - 477401)
 
gp: K = bnfinit(x^15 - 5*x^14 - 80*x^13 + 290*x^12 + 2545*x^11 - 5909*x^10 - 39850*x^9 + 51015*x^8 + 314635*x^7 - 181935*x^6 - 1227567*x^5 + 144735*x^4 + 2166295*x^3 + 419510*x^2 - 1278880*x - 477401, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 80 x^{13} + 290 x^{12} + 2545 x^{11} - 5909 x^{10} - 39850 x^{9} + 51015 x^{8} + 314635 x^{7} - 181935 x^{6} - 1227567 x^{5} + 144735 x^{4} + 2166295 x^{3} + 419510 x^{2} - 1278880 x - 477401 \)

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:  \(286613963390460550785064697265625=5^{24}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $145.82$
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:  $5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(925=5^{2}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{925}(1,·)$, $\chi_{925}(676,·)$, $\chi_{925}(581,·)$, $\chi_{925}(396,·)$, $\chi_{925}(491,·)$, $\chi_{925}(556,·)$, $\chi_{925}(306,·)$, $\chi_{925}(211,·)$, $\chi_{925}(186,·)$, $\chi_{925}(371,·)$, $\chi_{925}(121,·)$, $\chi_{925}(26,·)$, $\chi_{925}(861,·)$, $\chi_{925}(766,·)$, $\chi_{925}(741,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{1}{7} a^{9} + \frac{2}{7} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{6} - \frac{2}{7} a^{4} - \frac{1}{7} a^{3} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{4} + \frac{1}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{15888071788681034422578466732462343} a^{14} + \frac{598014346113918083649368578180198}{15888071788681034422578466732462343} a^{13} + \frac{14122964037034759825947973103059}{369490041597233358664615505406101} a^{12} + \frac{7365516746922966260576008348955771}{15888071788681034422578466732462343} a^{11} - \frac{168895620913969203535392558060293}{15888071788681034422578466732462343} a^{10} + \frac{2340472191817985142232291597551925}{15888071788681034422578466732462343} a^{9} - \frac{1050342631625540666082074903941747}{15888071788681034422578466732462343} a^{8} - \frac{5903646795461607302331868764584799}{15888071788681034422578466732462343} a^{7} + \frac{3045141898258300489958937858899337}{15888071788681034422578466732462343} a^{6} - \frac{5669346460400241760409123122038873}{15888071788681034422578466732462343} a^{5} + \frac{869397659963546591479195615732349}{2269724541240147774654066676066049} a^{4} + \frac{81057743685468119224467463263894}{2269724541240147774654066676066049} a^{3} + \frac{680547287655413441117642082740638}{15888071788681034422578466732462343} a^{2} + \frac{4727932753570869230962888830649717}{15888071788681034422578466732462343} a + \frac{6189440195312424852556193420728498}{15888071788681034422578466732462343}$

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:  \( 234241175446.62338 \) (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

3.3.1369.1, 5.5.390625.1

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 $15$ $15$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ $15$ $15$ $15$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ R $15$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ $15$ $15$

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
$5$5.15.24.88$x^{15} + 375 x^{14} + 415 x^{13} + 575 x^{12} + 520 x^{11} + 378 x^{10} + 145 x^{9} + 275 x^{8} + 85 x^{7} + 545 x^{6} + 127 x^{5} + 380 x^{4} + 470 x^{3} + 615 x + 368$$5$$3$$24$$C_{15}$$[2]^{3}$
$37$37.15.10.1$x^{15} + 1975467 x^{6} - 1874161 x^{3} + 152348673529$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$