Properties

Label 15.3.59463924292145152.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{12}\cdot 13^{9}\cdot 37^{2}$
Root discriminant $13.13$
Ramified primes $2, 13, 37$
Class number $1$
Class group Trivial
Galois Group 15T56

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 8, -21, 27, -18, 10, -14, 24, -31, 30, -13, -9, 15, -6, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 6*x^13 + 15*x^12 - 9*x^11 - 13*x^10 + 30*x^9 - 31*x^8 + 24*x^7 - 14*x^6 + 10*x^5 - 18*x^4 + 27*x^3 - 21*x^2 + 8*x - 1)
gp: K = bnfinit(x^15 - x^14 - 6*x^13 + 15*x^12 - 9*x^11 - 13*x^10 + 30*x^9 - 31*x^8 + 24*x^7 - 14*x^6 + 10*x^5 - 18*x^4 + 27*x^3 - 21*x^2 + 8*x - 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut x^{14} \) \(\mathstrut -\mathstrut 6 x^{13} \) \(\mathstrut +\mathstrut 15 x^{12} \) \(\mathstrut -\mathstrut 9 x^{11} \) \(\mathstrut -\mathstrut 13 x^{10} \) \(\mathstrut +\mathstrut 30 x^{9} \) \(\mathstrut -\mathstrut 31 x^{8} \) \(\mathstrut +\mathstrut 24 x^{7} \) \(\mathstrut -\mathstrut 14 x^{6} \) \(\mathstrut +\mathstrut 10 x^{5} \) \(\mathstrut -\mathstrut 18 x^{4} \) \(\mathstrut +\mathstrut 27 x^{3} \) \(\mathstrut -\mathstrut 21 x^{2} \) \(\mathstrut +\mathstrut 8 x \) \(\mathstrut -\mathstrut 1 \)

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:  \(59463924292145152=2^{12}\cdot 13^{9}\cdot 37^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $13.13$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 13, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2654} a^{14} - \frac{365}{2654} a^{13} + \frac{77}{1327} a^{12} - \frac{307}{2654} a^{11} + \frac{271}{2654} a^{10} - \frac{459}{2654} a^{9} + \frac{1231}{2654} a^{8} - \frac{458}{1327} a^{7} + \frac{371}{2654} a^{6} + \frac{148}{1327} a^{5} + \frac{540}{1327} a^{4} + \frac{981}{2654} a^{3} - \frac{47}{1327} a^{2} + \frac{510}{1327} a + \frac{144}{1327}$

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:  \( \frac{4997}{2654} a^{14} + \frac{2047}{2654} a^{13} - \frac{15985}{1327} a^{12} + \frac{29127}{2654} a^{11} + \frac{27187}{2654} a^{10} - \frac{31468}{1327} a^{9} + \frac{22890}{1327} a^{8} - \frac{26969}{2654} a^{7} + \frac{19973}{2654} a^{6} + \frac{1744}{1327} a^{5} + \frac{21083}{2654} a^{4} - \frac{25817}{1327} a^{3} + \frac{35869}{2654} a^{2} + \frac{1957}{1327} a - \frac{5967}{2654} \),  \( \frac{3480}{1327} a^{14} - \frac{4503}{2654} a^{13} - \frac{22747}{1327} a^{12} + \frac{43666}{1327} a^{11} - \frac{18085}{2654} a^{10} - \frac{54019}{1327} a^{9} + \frac{158561}{2654} a^{8} - \frac{67903}{1327} a^{7} + \frac{50335}{1327} a^{6} - \frac{48443}{2654} a^{5} + \frac{44463}{2654} a^{4} - \frac{111123}{2654} a^{3} + \frac{68326}{1327} a^{2} - \frac{38608}{1327} a + \frac{8317}{1327} \),  \( \frac{940}{1327} a^{14} - \frac{141}{2654} a^{13} - \frac{11709}{2654} a^{12} + \frac{8668}{1327} a^{11} + \frac{1239}{2654} a^{10} - \frac{25583}{2654} a^{9} + \frac{30513}{2654} a^{8} - \frac{24847}{2654} a^{7} + \frac{9028}{1327} a^{6} - \frac{7495}{2654} a^{5} + \frac{6680}{1327} a^{4} - \frac{13395}{1327} a^{3} + \frac{26311}{2654} a^{2} - \frac{4602}{1327} a + \frac{1339}{1327} \),  \( \frac{12}{1327} a^{14} + \frac{3183}{2654} a^{13} + \frac{521}{1327} a^{12} - \frac{19311}{2654} a^{11} + \frac{9887}{1327} a^{10} + \frac{5108}{1327} a^{9} - \frac{18403}{1327} a^{8} + \frac{37731}{2654} a^{7} - \frac{26925}{2654} a^{6} + \frac{8860}{1327} a^{5} - \frac{1947}{2654} a^{4} + \frac{7791}{1327} a^{3} - \frac{15725}{1327} a^{2} + \frac{14894}{1327} a - \frac{7685}{2654} \),  \( \frac{2295}{2654} a^{14} - \frac{169}{1327} a^{13} - \frac{16803}{2654} a^{12} + \frac{19977}{2654} a^{11} + \frac{14179}{2654} a^{10} - \frac{39577}{2654} a^{9} + \frac{27829}{2654} a^{8} - \frac{12195}{2654} a^{7} + \frac{7473}{2654} a^{6} + \frac{2602}{1327} a^{5} + \frac{3745}{2654} a^{4} - \frac{16186}{1327} a^{3} + \frac{21803}{2654} a^{2} + \frac{4053}{2654} a - \frac{3924}{1327} \),  \( \frac{720}{1327} a^{14} - \frac{54}{1327} a^{13} - \frac{10465}{2654} a^{12} + \frac{5877}{1327} a^{11} + \frac{9391}{2654} a^{10} - \frac{22673}{2654} a^{9} + \frac{7846}{1327} a^{8} - \frac{3982}{1327} a^{7} + \frac{3047}{1327} a^{6} + \frac{273}{2654} a^{5} + \frac{3937}{2654} a^{4} - \frac{19193}{2654} a^{3} + \frac{5305}{1327} a^{2} + \frac{2465}{2654} a - \frac{3285}{2654} \),  \( \frac{601}{2654} a^{14} - \frac{205}{1327} a^{13} - \frac{4317}{2654} a^{12} + \frac{3954}{1327} a^{11} + \frac{977}{2654} a^{10} - \frac{7220}{1327} a^{9} + \frac{12635}{2654} a^{8} - \frac{1896}{1327} a^{7} + \frac{2008}{1327} a^{6} + \frac{39}{1327} a^{5} - \frac{575}{1327} a^{4} - \frac{4448}{1327} a^{3} + \frac{11183}{2654} a^{2} - \frac{27}{1327} a - \frac{3403}{2654} \),  \( \frac{3480}{1327} a^{14} - \frac{4503}{2654} a^{13} - \frac{22747}{1327} a^{12} + \frac{43666}{1327} a^{11} - \frac{18085}{2654} a^{10} - \frac{54019}{1327} a^{9} + \frac{158561}{2654} a^{8} - \frac{67903}{1327} a^{7} + \frac{50335}{1327} a^{6} - \frac{48443}{2654} a^{5} + \frac{44463}{2654} a^{4} - \frac{111123}{2654} a^{3} + \frac{68326}{1327} a^{2} - \frac{37281}{1327} a + \frac{8317}{1327} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 226.669584518 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T56:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 4860
The 63 conjugacy class representatives for [3^5]F(5)=3wrF(5) are not computed
Character table for [3^5]F(5)=3wrF(5) is not computed

Intermediate fields

5.1.35152.1

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

Sibling fields

Degree 15 sibling: 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 R $15$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ R ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.12.9.2$x^{12} - 52 x^{8} + 676 x^{4} - 79092$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$37$37.3.2.2$x^{3} + 74$$3$$1$$2$$C_3$$[\ ]_{3}$
37.12.0.1$x^{12} - x + 15$$1$$12$$0$$C_{12}$$[\ ]^{12}$