Properties

Label 15.3.98765528526263521.1
Degree $15$
Signature $[3, 6]$
Discriminant $13^{3}\cdot 31^{6}\cdot 37^{3}$
Root discriminant $13.58$
Ramified primes $13, 31, 37$
Class number $1$
Class group Trivial
Galois Group $S_5 \times S_3$ (as 15T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -5, 8, -17, 30, -41, 54, -49, 36, -17, -2, 8, -12, 7, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 + 7*x^13 - 12*x^12 + 8*x^11 - 2*x^10 - 17*x^9 + 36*x^8 - 49*x^7 + 54*x^6 - 41*x^5 + 30*x^4 - 17*x^3 + 8*x^2 - 5*x + 3)
gp: K = bnfinit(x^15 - 3*x^14 + 7*x^13 - 12*x^12 + 8*x^11 - 2*x^10 - 17*x^9 + 36*x^8 - 49*x^7 + 54*x^6 - 41*x^5 + 30*x^4 - 17*x^3 + 8*x^2 - 5*x + 3, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 3 x^{14} \) \(\mathstrut +\mathstrut 7 x^{13} \) \(\mathstrut -\mathstrut 12 x^{12} \) \(\mathstrut +\mathstrut 8 x^{11} \) \(\mathstrut -\mathstrut 2 x^{10} \) \(\mathstrut -\mathstrut 17 x^{9} \) \(\mathstrut +\mathstrut 36 x^{8} \) \(\mathstrut -\mathstrut 49 x^{7} \) \(\mathstrut +\mathstrut 54 x^{6} \) \(\mathstrut -\mathstrut 41 x^{5} \) \(\mathstrut +\mathstrut 30 x^{4} \) \(\mathstrut -\mathstrut 17 x^{3} \) \(\mathstrut +\mathstrut 8 x^{2} \) \(\mathstrut -\mathstrut 5 x \) \(\mathstrut +\mathstrut 3 \)

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:  \(98765528526263521=13^{3}\cdot 31^{6}\cdot 37^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $13.58$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $13, 31, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{12437511} a^{14} - \frac{1221304}{12437511} a^{13} - \frac{1764504}{4145837} a^{12} - \frac{5260330}{12437511} a^{11} - \frac{255860}{4145837} a^{10} - \frac{716396}{4145837} a^{9} + \frac{5569642}{12437511} a^{8} + \frac{2234315}{12437511} a^{7} + \frac{2186188}{12437511} a^{6} - \frac{4083305}{12437511} a^{5} - \frac{4076633}{12437511} a^{4} - \frac{2009043}{4145837} a^{3} + \frac{5081312}{12437511} a^{2} - \frac{1125285}{4145837} a + \frac{1278925}{4145837}$

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:  \( \frac{764082}{4145837} a^{14} - \frac{2390109}{4145837} a^{13} + \frac{5487053}{4145837} a^{12} - \frac{9120626}{4145837} a^{11} + \frac{5061319}{4145837} a^{10} + \frac{1878610}{4145837} a^{9} - \frac{17752671}{4145837} a^{8} + \frac{31259807}{4145837} a^{7} - \frac{34473809}{4145837} a^{6} + \frac{31970895}{4145837} a^{5} - \frac{15057718}{4145837} a^{4} + \frac{8125718}{4145837} a^{3} - \frac{2383}{4145837} a^{2} + \frac{1949528}{4145837} a - \frac{1835564}{4145837} \),  \( \frac{9923}{4145837} a^{14} + \frac{1991714}{12437511} a^{13} - \frac{3440195}{12437511} a^{12} + \frac{1979077}{4145837} a^{11} - \frac{6527150}{12437511} a^{10} - \frac{4352833}{4145837} a^{9} + \frac{3550356}{4145837} a^{8} - \frac{23214778}{12437511} a^{7} + \frac{15864694}{12437511} a^{6} + \frac{34295}{12437511} a^{5} - \frac{347113}{12437511} a^{4} + \frac{6076322}{12437511} a^{3} + \frac{4335219}{4145837} a^{2} - \frac{17321963}{12437511} a + \frac{1097154}{4145837} \),  \( \frac{43037}{12437511} a^{14} - \frac{338762}{12437511} a^{13} + \frac{337681}{4145837} a^{12} - \frac{1246988}{12437511} a^{11} - \frac{103748}{4145837} a^{10} + \frac{1055117}{4145837} a^{9} - \frac{7466749}{12437511} a^{8} + \frac{3817114}{12437511} a^{7} + \frac{9639752}{12437511} a^{6} - \frac{3604366}{12437511} a^{5} + \frac{9913256}{12437511} a^{4} - \frac{1752956}{4145837} a^{3} - \frac{16768880}{12437511} a^{2} + \frac{2777289}{4145837} a - \frac{3182624}{4145837} \),  \( \frac{4108882}{12437511} a^{14} - \frac{3576591}{4145837} a^{13} + \frac{21146300}{12437511} a^{12} - \frac{31822639}{12437511} a^{11} + \frac{102146}{12437511} a^{10} + \frac{7390772}{4145837} a^{9} - \frac{68229800}{12437511} a^{8} + \frac{35723543}{4145837} a^{7} - \frac{29238166}{4145837} a^{6} + \frac{93715400}{12437511} a^{5} - \frac{28473250}{12437511} a^{4} + \frac{22334392}{12437511} a^{3} - \frac{27364741}{12437511} a^{2} - \frac{1805479}{12437511} a - \frac{4277412}{4145837} \),  \( \frac{3084457}{12437511} a^{14} - \frac{7069433}{12437511} a^{13} + \frac{14339182}{12437511} a^{12} - \frac{18653359}{12437511} a^{11} - \frac{4368470}{12437511} a^{10} + \frac{5297332}{4145837} a^{9} - \frac{53001911}{12437511} a^{8} + \frac{52717714}{12437511} a^{7} - \frac{71479813}{12437511} a^{6} + \frac{11219023}{4145837} a^{5} - \frac{2746355}{4145837} a^{4} + \frac{18666161}{12437511} a^{3} + \frac{8630978}{12437511} a^{2} + \frac{12915139}{12437511} a + \frac{388203}{4145837} \),  \( \frac{3430810}{12437511} a^{14} - \frac{1826266}{4145837} a^{13} + \frac{12365630}{12437511} a^{12} - \frac{16055992}{12437511} a^{11} - \frac{10352422}{12437511} a^{10} - \frac{553680}{4145837} a^{9} - \frac{48742385}{12437511} a^{8} + \frac{14731605}{4145837} a^{7} - \frac{17882648}{4145837} a^{6} + \frac{54679562}{12437511} a^{5} - \frac{2163913}{12437511} a^{4} + \frac{32885212}{12437511} a^{3} + \frac{3604592}{12437511} a^{2} + \frac{9234086}{12437511} a - \frac{2055537}{4145837} \),  \( \frac{161350}{4145837} a^{14} - \frac{720022}{12437511} a^{13} - \frac{2363261}{12437511} a^{12} + \frac{2234325}{4145837} a^{11} - \frac{17916245}{12437511} a^{10} + \frac{7052065}{4145837} a^{9} + \frac{1816906}{4145837} a^{8} - \frac{16759951}{12437511} a^{7} + \frac{49157194}{12437511} a^{6} - \frac{62328892}{12437511} a^{5} + \frac{57874958}{12437511} a^{4} - \frac{35316409}{12437511} a^{3} + \frac{1403591}{4145837} a^{2} - \frac{9250874}{12437511} a + \frac{3119573}{4145837} \),  \( \frac{1325769}{4145837} a^{14} - \frac{14443930}{12437511} a^{13} + \frac{30955048}{12437511} a^{12} - \frac{17130013}{4145837} a^{11} + \frac{30872185}{12437511} a^{10} + \frac{7444440}{4145837} a^{9} - \frac{29388447}{4145837} a^{8} + \frac{170663729}{12437511} a^{7} - \frac{188437952}{12437511} a^{6} + \frac{174324848}{12437511} a^{5} - \frac{116794075}{12437511} a^{4} + \frac{46860392}{12437511} a^{3} - \frac{15820786}{4145837} a^{2} + \frac{12069151}{12437511} a - \frac{3456294}{4145837} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 363.728980494 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_5 \times S_3$ (as 15T29):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 720
The 21 conjugacy class representatives for $S_5 \times S_3$
Character table for $S_5 \times S_3$ is not computed

Intermediate fields

3.1.31.1, 5.3.14911.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: 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 $15$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ R R ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $15$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
31Data not computed
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$