Properties

Label 15.1.44543599279432079.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,239^{7}$
Root discriminant $12.88$
Ramified prime $239$
Class number $1$
Class group Trivial
Galois Group $D_{15}$ (as 15T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 6, -2, -8, -4, 27, -13, -15, 4, 20, -13, -5, 4, 4, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 4*x^13 + 4*x^12 - 5*x^11 - 13*x^10 + 20*x^9 + 4*x^8 - 15*x^7 - 13*x^6 + 27*x^5 - 4*x^4 - 8*x^3 - 2*x^2 + 6*x - 1)
gp: K = bnfinit(x^15 - 4*x^14 + 4*x^13 + 4*x^12 - 5*x^11 - 13*x^10 + 20*x^9 + 4*x^8 - 15*x^7 - 13*x^6 + 27*x^5 - 4*x^4 - 8*x^3 - 2*x^2 + 6*x - 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 4 x^{14} \) \(\mathstrut +\mathstrut 4 x^{13} \) \(\mathstrut +\mathstrut 4 x^{12} \) \(\mathstrut -\mathstrut 5 x^{11} \) \(\mathstrut -\mathstrut 13 x^{10} \) \(\mathstrut +\mathstrut 20 x^{9} \) \(\mathstrut +\mathstrut 4 x^{8} \) \(\mathstrut -\mathstrut 15 x^{7} \) \(\mathstrut -\mathstrut 13 x^{6} \) \(\mathstrut +\mathstrut 27 x^{5} \) \(\mathstrut -\mathstrut 4 x^{4} \) \(\mathstrut -\mathstrut 8 x^{3} \) \(\mathstrut -\mathstrut 2 x^{2} \) \(\mathstrut +\mathstrut 6 x \) \(\mathstrut -\mathstrut 1 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[1, 7]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-44543599279432079=-\,239^{7}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $12.88$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{34333} a^{14} - \frac{6031}{34333} a^{13} - \frac{9806}{34333} a^{12} + \frac{13673}{34333} a^{11} - \frac{7976}{34333} a^{10} + \frac{5139}{34333} a^{9} - \frac{4367}{34333} a^{8} - \frac{13498}{34333} a^{7} - \frac{16779}{34333} a^{6} + \frac{16335}{34333} a^{5} + \frac{16026}{34333} a^{4} - \frac{9977}{34333} a^{3} + \frac{752}{1807} a^{2} - \frac{6614}{34333} a + \frac{1971}{34333}$

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:  $7$
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:  \( \frac{2825}{34333} a^{14} - \frac{8407}{34333} a^{13} + \frac{4781}{34333} a^{12} + \frac{1600}{34333} a^{11} + \frac{24581}{34333} a^{10} - \frac{39517}{34333} a^{9} - \frac{11228}{34333} a^{8} + \frac{12113}{34333} a^{7} + \frac{81864}{34333} a^{6} - \frac{65843}{34333} a^{5} - \frac{46110}{34333} a^{4} + \frac{2368}{34333} a^{3} + \frac{4789}{1807} a^{2} - \frac{41731}{34333} a - \frac{28204}{34333} \),  \( \frac{9151}{34333} a^{14} - \frac{16550}{34333} a^{13} - \frac{22577}{34333} a^{12} + \frac{46504}{34333} a^{11} + \frac{72248}{34333} a^{10} - \frac{112220}{34333} a^{9} - \frac{101804}{34333} a^{8} + \frac{147268}{34333} a^{7} + \frac{129879}{34333} a^{6} - \frac{175962}{34333} a^{5} - \frac{119649}{34333} a^{4} + \frac{129252}{34333} a^{3} + \frac{4110}{1807} a^{2} - \frac{29968}{34333} a - \frac{22537}{34333} \),  \( \frac{4072}{34333} a^{14} - \frac{10137}{34333} a^{13} - \frac{753}{34333} a^{12} + \frac{22663}{34333} a^{11} + \frac{746}{34333} a^{10} - \frac{51455}{34333} a^{9} + \frac{36403}{34333} a^{8} + \frac{37610}{34333} a^{7} - \frac{35751}{34333} a^{6} - \frac{55567}{34333} a^{5} + \frac{59505}{34333} a^{4} - \frac{10405}{34333} a^{3} - \frac{721}{1807} a^{2} - \frac{15136}{34333} a + \frac{26323}{34333} \),  \( \frac{1179}{34333} a^{14} - \frac{3618}{34333} a^{13} + \frac{8947}{34333} a^{12} - \frac{16043}{34333} a^{11} + \frac{3538}{34333} a^{10} + \frac{16273}{34333} a^{9} + \frac{35590}{34333} a^{8} - \frac{86629}{34333} a^{7} - \frac{6633}{34333} a^{6} + \frac{66818}{34333} a^{5} + \frac{45837}{34333} a^{4} - \frac{123996}{34333} a^{3} + \frac{1178}{1807} a^{2} + \frac{30018}{34333} a + \frac{23498}{34333} \),  \( \frac{1744}{34333} a^{14} - \frac{12166}{34333} a^{13} + \frac{30503}{34333} a^{12} - \frac{15723}{34333} a^{11} - \frac{39612}{34333} a^{10} + \frac{1503}{34333} a^{9} + \frac{143210}{34333} a^{8} - \frac{91073}{34333} a^{7} - \frac{113859}{34333} a^{6} + \frac{26183}{34333} a^{5} + \frac{208280}{34333} a^{4} - \frac{130389}{34333} a^{3} - \frac{4008}{1807} a^{2} + \frac{1072}{34333} a + \frac{72790}{34333} \),  \( \frac{4567}{34333} a^{14} - \frac{8511}{34333} a^{13} - \frac{13770}{34333} a^{12} + \frac{27197}{34333} a^{11} + \frac{35254}{34333} a^{10} - \frac{48292}{34333} a^{9} - \frac{65282}{34333} a^{8} + \frac{51035}{34333} a^{7} + \frac{70229}{34333} a^{6} - \frac{37997}{34333} a^{5} - \frac{75880}{34333} a^{4} + \frac{29265}{34333} a^{3} + \frac{2891}{1807} a^{2} + \frac{6902}{34333} a - \frac{28022}{34333} \),  \( \frac{13063}{34333} a^{14} - \frac{57384}{34333} a^{13} + \frac{69311}{34333} a^{12} + \frac{44466}{34333} a^{11} - \frac{92832}{34333} a^{10} - \frac{161923}{34333} a^{9} + \frac{324322}{34333} a^{8} + \frac{9914}{34333} a^{7} - \frac{242536}{34333} a^{6} - \frac{132822}{34333} a^{5} + \frac{397000}{34333} a^{4} - \frac{104482}{34333} a^{3} - \frac{4897}{1807} a^{2} - \frac{16854}{34333} a + \frac{31756}{34333} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 124.657592501 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_{15}$ (as 15T2):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.239.1, 5.1.57121.1

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

Sibling fields

Galois closure: 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 $15$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
239Data not computed