Properties

Label 15.3.21643467887730481.1
Degree $15$
Signature $[3, 6]$
Discriminant $13^{2}\cdot 47^{6}\cdot 109^{2}$
Root discriminant $12.28$
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
Root discriminant:  $12.28$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
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 $\Q$.
This is not a CM field.

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

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

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
The 72 conjugacy class representatives for [3^5]D(5)=3wrD(5) are not computed
Character table for [3^5]D(5)=3wrD(5) is not computed

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
Degree 30 siblings: data not computed
Degree 45 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}$