# Properties

 Label 15.3.21643467887730481.1 Degree 15 Signature $[3, 6]$ Discriminant $13^{2}\cdot 47^{6}\cdot 109^{2}$ Ramified primes $13, 47, 109$ Class number 1 Class group Trivial Galois Group 15T46

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: K<a> := NumberField(PolynomialRing(Rationals())![-1, -2, -11, 22, -21, -11, 7, -17, 12, 18, -2, -7, 1, -1, -1, 1]);
sage: K = NumberField(x^15 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 2*x - 1,"a")
gp: K = bnfinit(x^15 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 2*x - 1, 1)

## Normalizeddefining polynomial

$x^{15}$ $\mathstrut -\mathstrut x^{14}$ $\mathstrut -\mathstrut x^{13}$ $\mathstrut +\mathstrut x^{12}$ $\mathstrut -\mathstrut 7 x^{11}$ $\mathstrut -\mathstrut 2 x^{10}$ $\mathstrut +\mathstrut 18 x^{9}$ $\mathstrut +\mathstrut 12 x^{8}$ $\mathstrut -\mathstrut 17 x^{7}$ $\mathstrut +\mathstrut 7 x^{6}$ $\mathstrut -\mathstrut 11 x^{5}$ $\mathstrut -\mathstrut 21 x^{4}$ $\mathstrut +\mathstrut 22 x^{3}$ $\mathstrut -\mathstrut 11 x^{2}$ $\mathstrut -\mathstrut 2 x$ $\mathstrut -\mathstrut 1$

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

## Invariants

 Degree: $15$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[3, 6]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $21643467887730481=13^{2}\cdot 47^{6}\cdot 109^{2}$ magma: Discriminant(K); sage: K.disc() gp: K.disc Ramified primes: $13, 47, 109$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is not Galois over $\Q$.

## Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4823316529} a^{14} - \frac{240451914}{4823316529} a^{13} - \frac{1467210693}{4823316529} a^{12} + \frac{2044776038}{4823316529} a^{11} + \frac{2380646120}{4823316529} a^{10} - \frac{2290615831}{4823316529} a^{9} - \frac{1561717558}{4823316529} a^{8} + \frac{2080171999}{4823316529} a^{7} - \frac{2386498101}{4823316529} a^{6} + \frac{1715950732}{4823316529} a^{5} + \frac{1834453739}{4823316529} a^{4} + \frac{64937028}{4823316529} a^{3} + \frac{593205418}{4823316529} a^{2} + \frac{970975456}{4823316529} a + \frac{820723294}{4823316529}$

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

## Class group and class number

Trivial Abelian group, order 1

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
 Rank: $8$ magma: UnitRank(K); sage: UK.rank() gp: #K.fu Torsion generator: $-1$ magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $\frac{438282190}{4823316529} a^{14} - \frac{666156758}{4823316529} a^{13} - \frac{180199038}{4823316529} a^{12} + \frac{937506764}{4823316529} a^{11} - \frac{3215866687}{4823316529} a^{10} + \frac{764978875}{4823316529} a^{9} + \frac{8142107838}{4823316529} a^{8} - \frac{819825977}{4823316529} a^{7} - \frac{12445039424}{4823316529} a^{6} + \frac{7101345595}{4823316529} a^{5} - \frac{2075689344}{4823316529} a^{4} - \frac{3242761291}{4823316529} a^{3} + \frac{18275305195}{4823316529} a^{2} - \frac{11471590121}{4823316529} a - \frac{1755151205}{4823316529}$,  $\frac{279318677}{4823316529} a^{14} - \frac{570733139}{4823316529} a^{13} - \frac{97673519}{4823316529} a^{12} + \frac{286289673}{4823316529} a^{11} - \frac{2256835527}{4823316529} a^{10} + \frac{1369023982}{4823316529} a^{9} + \frac{6081306034}{4823316529} a^{8} + \frac{1059190063}{4823316529} a^{7} - \frac{6065350387}{4823316529} a^{6} + \frac{5660375798}{4823316529} a^{5} - \frac{6997088601}{4823316529} a^{4} - \frac{5291197834}{4823316529} a^{3} + \frac{4910463282}{4823316529} a^{2} - \frac{11751794763}{4823316529} a + \frac{4235471514}{4823316529}$,  $\frac{38028855}{4823316529} a^{14} + \frac{30131136}{4823316529} a^{13} - \frac{41830710}{4823316529} a^{12} + \frac{95826109}{4823316529} a^{11} + \frac{13051036}{4823316529} a^{10} - \frac{448791598}{4823316529} a^{9} + \frac{291200427}{4823316529} a^{8} + \frac{903950089}{4823316529} a^{7} - \frac{1148631260}{4823316529} a^{6} - \frac{1349674114}{4823316529} a^{5} + \frac{1574584533}{4823316529} a^{4} + \frac{638893288}{4823316529} a^{3} - \frac{3508838763}{4823316529} a^{2} + \frac{1850995278}{4823316529} a + \frac{1978357973}{4823316529}$,  $\frac{566139539}{4823316529} a^{14} - \frac{325368897}{4823316529} a^{13} - \frac{420696872}{4823316529} a^{12} + \frac{578463500}{4823316529} a^{11} - \frac{3451205678}{4823316529} a^{10} - \frac{2747557933}{4823316529} a^{9} + \frac{7365442564}{4823316529} a^{8} + \frac{6303751499}{4823316529} a^{7} - \frac{9186881148}{4823316529} a^{6} + \frac{2757048777}{4823316529} a^{5} - \frac{662352361}{4823316529} a^{4} - \frac{5601051546}{4823316529} a^{3} + \frac{12324635935}{4823316529} a^{2} + \frac{209859584}{4823316529} a - \frac{2375337177}{4823316529}$,  $\frac{469621379}{4823316529} a^{14} + \frac{58489305}{4823316529} a^{13} - \frac{825121412}{4823316529} a^{12} + \frac{90755217}{4823316529} a^{11} - \frac{2804712262}{4823316529} a^{10} - \frac{4403499472}{4823316529} a^{9} + \frac{6154418487}{4823316529} a^{8} + \frac{12709698685}{4823316529} a^{7} - \frac{2583762033}{4823316529} a^{6} - \frac{4750900235}{4823316529} a^{5} - \frac{1059112278}{4823316529} a^{4} - \frac{12098985853}{4823316529} a^{3} + \frac{4091725504}{4823316529} a^{2} + \frac{10667639529}{4823316529} a - \frac{2580378452}{4823316529}$,  $\frac{1205387739}{4823316529} a^{14} - \frac{816773200}{4823316529} a^{13} - \frac{1373326882}{4823316529} a^{12} + \frac{706262859}{4823316529} a^{11} - \frac{7858948337}{4823316529} a^{10} - \frac{4862149836}{4823316529} a^{9} + \frac{19365056971}{4823316529} a^{8} + \frac{20389332875}{4823316529} a^{7} - \frac{14912893927}{4823316529} a^{6} + \frac{1147428040}{4823316529} a^{5} - \frac{12087988177}{4823316529} a^{4} - \frac{21912285059}{4823316529} a^{3} + \frac{18575332517}{4823316529} a^{2} - \frac{6222009777}{4823316529} a + \frac{700795973}{4823316529}$,  $\frac{667132927}{4823316529} a^{14} - \frac{753110120}{4823316529} a^{13} - \frac{1017644411}{4823316529} a^{12} + \frac{516490886}{4823316529} a^{11} - \frac{4405300094}{4823316529} a^{10} - \frac{813451459}{4823316529} a^{9} + \frac{14786449294}{4823316529} a^{8} + \frac{11452381693}{4823316529} a^{7} - \frac{13822873461}{4823316529} a^{6} - \frac{3214737208}{4823316529} a^{5} - \frac{9422680395}{4823316529} a^{4} - \frac{15211313221}{4823316529} a^{3} + \frac{15929636186}{4823316529} a^{2} - \frac{4384209022}{4823316529} a - \frac{4901669196}{4823316529}$,  $\frac{648967434}{4823316529} a^{14} - \frac{1114737512}{4823316529} a^{13} - \frac{747311498}{4823316529} a^{12} + \frac{1124779681}{4823316529} a^{11} - \frac{4823766233}{4823316529} a^{10} + \frac{1779201762}{4823316529} a^{9} + \frac{15799224682}{4823316529} a^{8} + \frac{4012111589}{4823316529} a^{7} - \frac{20004131378}{4823316529} a^{6} + \frac{3648655910}{4823316529} a^{5} - \frac{7466582441}{4823316529} a^{4} - \frac{13931973140}{4823316529} a^{3} + \frac{24631797650}{4823316529} a^{2} - \frac{10717944600}{4823316529} a - \frac{879206783}{4823316529}$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $116.633322592$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 2430 Conjugacy class representatives for 15T46 Character table for 15T46

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Sibling fields

 Degree 15 siblings: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 $15$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R $15$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ $15$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ R $15$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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$$[\ ]_{3} 13.6.0.1x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6} 4747.3.0.1x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3} 47.6.3.2x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$109$109.3.2.2$x^{3} + 654$$3$$1$$2$$C_3$$[\ ]_{3} 109.6.0.1x^{6} - x + 11$$1$$6$$0$$C_6$$[\ ]^{6}$
109.6.0.1$x^{6} - x + 11$$1$$6$$0$$C_6$$[\ ]^{6}$