Properties

Label 15.1.69378727128301847.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,23^{5}\cdot 47^{6}$
Root discriminant $13.27$
Ramified primes $23, 47$
Class number $1$
Class group Trivial
Galois Group $D_5\times S_3$ (as 15T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 2, 6, 3, 3, -5, 1, 2, 3, -4, 2, -2, 2, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 2*x^13 - 2*x^12 + 2*x^11 - 4*x^10 + 3*x^9 + 2*x^8 + x^7 - 5*x^6 + 3*x^5 + 3*x^4 + 6*x^3 + 2*x^2 - 1)
gp: K = bnfinit(x^15 - 2*x^14 + 2*x^13 - 2*x^12 + 2*x^11 - 4*x^10 + 3*x^9 + 2*x^8 + x^7 - 5*x^6 + 3*x^5 + 3*x^4 + 6*x^3 + 2*x^2 - 1, 1)

Normalized defining polynomial

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

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

Invariants

Degree:  $15$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[1, 7]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-69378727128301847=-\,23^{5}\cdot 47^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $13.27$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $23, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5995} a^{14} - \frac{258}{5995} a^{13} + \frac{21}{1199} a^{12} - \frac{2902}{5995} a^{11} - \frac{466}{5995} a^{10} - \frac{608}{5995} a^{9} - \frac{219}{5995} a^{8} + \frac{2111}{5995} a^{7} - \frac{173}{1199} a^{6} - \frac{76}{1199} a^{5} + \frac{1363}{5995} a^{4} - \frac{243}{1199} a^{3} - \frac{694}{5995} a^{2} - \frac{2184}{5995} a + \frac{1569}{5995}$

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:  \( \frac{1982}{5995} a^{14} - \frac{596}{1199} a^{13} + \frac{683}{5995} a^{12} + \frac{2237}{5995} a^{11} - \frac{3979}{5995} a^{10} - \frac{61}{5995} a^{9} - \frac{3617}{5995} a^{8} + \frac{15079}{5995} a^{7} - \frac{7054}{5995} a^{6} - \frac{6183}{5995} a^{5} + \frac{3716}{5995} a^{4} + \frac{15049}{5995} a^{3} + \frac{4541}{5995} a^{2} + \frac{9294}{5995} a - \frac{1647}{5995} \),  \( \frac{1006}{5995} a^{14} - \frac{2962}{5995} a^{13} + \frac{2516}{5995} a^{12} + \frac{153}{5995} a^{11} - \frac{477}{1199} a^{10} + \frac{1041}{5995} a^{9} + \frac{302}{5995} a^{8} + \frac{7431}{5995} a^{7} - \frac{9308}{5995} a^{6} - \frac{919}{1199} a^{5} + \frac{9114}{5995} a^{4} + \frac{1889}{5995} a^{3} - \frac{3943}{5995} a^{2} - \frac{347}{1199} a - \frac{1093}{1199} \),  \( \frac{1044}{5995} a^{14} - \frac{395}{1199} a^{13} + \frac{511}{5995} a^{12} + \frac{1384}{5995} a^{11} - \frac{2108}{5995} a^{10} + \frac{718}{5995} a^{9} - \frac{3224}{5995} a^{8} + \frac{10913}{5995} a^{7} - \frac{6208}{5995} a^{6} - \frac{5846}{5995} a^{5} + \frac{2157}{5995} a^{4} + \frac{10873}{5995} a^{3} + \frac{3257}{5995} a^{2} - \frac{797}{5995} a - \frac{4594}{5995} \),  \( \frac{2166}{5995} a^{14} - \frac{3691}{5995} a^{13} + \frac{2018}{5995} a^{12} - \frac{574}{5995} a^{11} + \frac{202}{5995} a^{10} - \frac{5222}{5995} a^{9} + \frac{2848}{5995} a^{8} + \frac{9032}{5995} a^{7} + \frac{1646}{5995} a^{6} - \frac{14954}{5995} a^{5} + \frac{9912}{5995} a^{4} + \frac{8508}{5995} a^{3} + \frac{12332}{5995} a^{2} + \frac{9103}{5995} a + \frac{488}{5995} \),  \( \frac{575}{1199} a^{14} - \frac{873}{1199} a^{13} - \frac{273}{5995} a^{12} + \frac{6586}{5995} a^{11} - \frac{11258}{5995} a^{10} + \frac{7336}{5995} a^{9} - \frac{2428}{1199} a^{8} + \frac{29762}{5995} a^{7} - \frac{15736}{5995} a^{6} - \frac{9803}{5995} a^{5} + \frac{5089}{5995} a^{4} + \frac{3989}{1199} a^{3} + \frac{3483}{5995} a^{2} + \frac{18148}{5995} a - \frac{8156}{5995} \),  \( \frac{387}{1199} a^{14} - \frac{6441}{5995} a^{13} + \frac{8937}{5995} a^{12} - \frac{8846}{5995} a^{11} + \frac{7132}{5995} a^{10} - \frac{1491}{1199} a^{9} + \frac{9074}{5995} a^{8} + \frac{4588}{5995} a^{7} - \frac{11961}{5995} a^{6} - \frac{1512}{5995} a^{5} + \frac{1120}{1199} a^{4} - \frac{2184}{5995} a^{3} + \frac{4786}{5995} a^{2} - \frac{3162}{5995} a - \frac{690}{1199} \),  \( \frac{174}{545} a^{14} - \frac{84}{109} a^{13} + \frac{612}{545} a^{12} - \frac{823}{545} a^{11} + \frac{993}{545} a^{10} - \frac{1479}{545} a^{9} + \frac{1461}{545} a^{8} - \frac{561}{545} a^{7} + \frac{564}{545} a^{6} - \frac{144}{109} a^{5} + \frac{414}{545} a^{4} + \frac{268}{545} a^{3} + \frac{1106}{545} a^{2} + \frac{67}{545} a + \frac{288}{545} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 185.949282778 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_3\times D_5$ (as 15T7):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 60
The 12 conjugacy class representatives for $D_5\times S_3$
Character table for $D_5\times S_3$

Intermediate fields

3.1.23.1, 5.1.2209.1

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

Sibling fields

Degree 30 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 $15$ $15$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ R ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$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}$