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

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, -11, 22, -21, -11, 7, -17, 12, 18, -2, -7, 1, -1, -1, 1]);
sage: x = polygen(QQ); K.<a> = 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)
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)

Normalized defining 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:  \( -1 \) (order $2$)
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

15T46:

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

5.1.2209.1

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $15$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ 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$$[\ ]_{3}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$47$47.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$109$109.3.2.2$x^{3} + 654$$3$$1$$2$$C_3$$[\ ]_{3}$
109.6.0.1$x^{6} - x + 11$$1$$6$$0$$C_6$$[\ ]^{6}$
109.6.0.1$x^{6} - x + 11$$1$$6$$0$$C_6$$[\ ]^{6}$