Properties

Label 15.3.137631064560062464.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{12}\cdot 23^{6}\cdot 61^{3}$
Root discriminant $13.89$
Ramified primes $2, 23, 61$
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![1, 0, -5, -13, 2, 14, 22, -10, -5, -14, -5, 11, 3, -2, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 2*x^13 + 3*x^12 + 11*x^11 - 5*x^10 - 14*x^9 - 5*x^8 - 10*x^7 + 22*x^6 + 14*x^5 + 2*x^4 - 13*x^3 - 5*x^2 + 1)
gp: K = bnfinit(x^15 - x^14 - 2*x^13 + 3*x^12 + 11*x^11 - 5*x^10 - 14*x^9 - 5*x^8 - 10*x^7 + 22*x^6 + 14*x^5 + 2*x^4 - 13*x^3 - 5*x^2 + 1, 1)

Normalized defining polynomial

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

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:  \(137631064560062464=2^{12}\cdot 23^{6}\cdot 61^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $13.89$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 23, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{23} a^{11} - \frac{6}{23} a^{10} - \frac{5}{23} a^{9} + \frac{1}{23} a^{8} + \frac{2}{23} a^{7} + \frac{2}{23} a^{6} - \frac{11}{23} a^{5} - \frac{7}{23} a^{4} + \frac{5}{23} a^{3} + \frac{10}{23} a^{2} + \frac{5}{23} a + \frac{1}{23}$, $\frac{1}{23} a^{12} + \frac{5}{23} a^{10} - \frac{6}{23} a^{9} + \frac{8}{23} a^{8} - \frac{9}{23} a^{7} + \frac{1}{23} a^{6} - \frac{4}{23} a^{5} + \frac{9}{23} a^{4} - \frac{6}{23} a^{3} - \frac{4}{23} a^{2} + \frac{8}{23} a + \frac{6}{23}$, $\frac{1}{529} a^{13} + \frac{8}{529} a^{12} + \frac{2}{529} a^{11} + \frac{98}{529} a^{10} + \frac{136}{529} a^{9} - \frac{201}{529} a^{8} - \frac{54}{529} a^{7} + \frac{67}{529} a^{6} + \frac{79}{529} a^{5} - \frac{212}{529} a^{4} + \frac{255}{529} a^{3} + \frac{199}{529} a^{2} - \frac{175}{529} a - \frac{1}{529}$, $\frac{1}{529} a^{14} + \frac{7}{529} a^{12} - \frac{10}{529} a^{11} + \frac{249}{529} a^{10} - \frac{185}{529} a^{9} - \frac{102}{529} a^{8} + \frac{223}{529} a^{7} - \frac{43}{529} a^{6} - \frac{108}{529} a^{5} + \frac{42}{529} a^{4} - \frac{70}{529} a^{3} + \frac{211}{529} a^{2} - \frac{96}{529} a - \frac{199}{529}$

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:  \( \frac{162}{529} a^{14} - \frac{84}{529} a^{13} - \frac{504}{529} a^{12} + \frac{466}{529} a^{11} + \frac{2114}{529} a^{10} - \frac{316}{529} a^{9} - \frac{3527}{529} a^{8} - \frac{1152}{529} a^{7} - \frac{1646}{529} a^{6} + \frac{2548}{529} a^{5} + \frac{5430}{529} a^{4} - \frac{882}{529} a^{3} - \frac{1624}{529} a^{2} - \frac{1542}{529} a + \frac{12}{23} \),  \( \frac{144}{529} a^{14} - \frac{44}{529} a^{13} - \frac{379}{529} a^{12} + \frac{358}{529} a^{11} + \frac{1828}{529} a^{10} + \frac{128}{529} a^{9} - \frac{2187}{529} a^{8} - \frac{564}{529} a^{7} - \frac{1642}{529} a^{6} + \frac{867}{529} a^{5} + \frac{2910}{529} a^{4} - \frac{370}{529} a^{3} - \frac{866}{529} a^{2} - \frac{213}{529} a + \frac{17}{23} \),  \( \frac{12}{529} a^{14} + \frac{71}{529} a^{13} - \frac{130}{529} a^{12} - \frac{1}{529} a^{11} + \frac{355}{529} a^{10} + \frac{605}{529} a^{9} - \frac{614}{529} a^{8} + \frac{15}{529} a^{7} - \frac{290}{529} a^{6} - \frac{1299}{529} a^{5} + \frac{1322}{529} a^{4} - \frac{1434}{529} a^{3} + \frac{515}{529} a^{2} - \frac{375}{529} a + \frac{761}{529} \),  \( \frac{150}{529} a^{14} - \frac{155}{529} a^{13} - \frac{374}{529} a^{12} + \frac{467}{529} a^{11} + \frac{1759}{529} a^{10} - \frac{921}{529} a^{9} - \frac{2913}{529} a^{8} - \frac{1167}{529} a^{7} - \frac{1356}{529} a^{6} + \frac{3847}{529} a^{5} + \frac{4108}{529} a^{4} + \frac{24}{23} a^{3} - \frac{93}{23} a^{2} - \frac{1167}{529} a - \frac{485}{529} \),  \( \frac{128}{529} a^{14} + \frac{133}{529} a^{13} - \frac{547}{529} a^{12} - \frac{48}{529} a^{11} + \frac{2241}{529} a^{10} + \frac{1975}{529} a^{9} - \frac{3334}{529} a^{8} - \frac{3340}{529} a^{7} - \frac{1929}{529} a^{6} - \frac{741}{529} a^{5} + \frac{7103}{529} a^{4} + \frac{109}{23} a^{3} - \frac{39}{23} a^{2} - \frac{2650}{529} a - \frac{535}{529} \),  \( \frac{208}{529} a^{14} - \frac{197}{529} a^{13} - \frac{350}{529} a^{12} + \frac{539}{529} a^{11} + \frac{2149}{529} a^{10} - \frac{665}{529} a^{9} - \frac{2135}{529} a^{8} - \frac{1536}{529} a^{7} - \frac{3122}{529} a^{6} + \frac{4339}{529} a^{5} + \frac{1947}{529} a^{4} + \frac{1376}{529} a^{3} - \frac{1295}{529} a^{2} - \frac{1363}{529} a + \frac{113}{529} \),  \( \frac{68}{529} a^{14} + \frac{141}{529} a^{13} - \frac{213}{529} a^{12} - \frac{145}{529} a^{11} + \frac{1103}{529} a^{10} + \frac{1950}{529} a^{9} - \frac{363}{529} a^{8} - \frac{1512}{529} a^{7} - \frac{2194}{529} a^{6} - \frac{123}{23} a^{5} + \frac{1392}{529} a^{4} + \frac{1571}{529} a^{3} + \frac{1950}{529} a^{2} - \frac{38}{529} a + \frac{12}{529} \),  \( \frac{164}{529} a^{14} - \frac{234}{529} a^{13} - \frac{287}{529} a^{12} + \frac{721}{529} a^{11} + \frac{1528}{529} a^{10} - \frac{1697}{529} a^{9} - \frac{1986}{529} a^{8} + \frac{678}{529} a^{7} - \frac{1823}{529} a^{6} + \frac{4466}{529} a^{5} + \frac{1480}{529} a^{4} - \frac{2495}{529} a^{3} - \frac{1290}{529} a^{2} - \frac{2}{529} a + \frac{557}{529} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 381.090214491 \)
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

3.1.23.1, 5.3.22448.1

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 R $15$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
2Data not computed
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
61Data not computed