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Properties

Label 15.15.4010958493...8161.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{19}\cdot 83\cdot 401^{6}$
Root discriminant $59.37$
Ramified primes $3, 83, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T86

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![243, 486, -4617, 3483, 10260, -11826, -7125, 10989, 1809, -4259, -111, 720, -4, -48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 48*x^13 - 4*x^12 + 720*x^11 - 111*x^10 - 4259*x^9 + 1809*x^8 + 10989*x^7 - 7125*x^6 - 11826*x^5 + 10260*x^4 + 3483*x^3 - 4617*x^2 + 486*x + 243)
 
gp: K = bnfinit(x^15 - 48*x^13 - 4*x^12 + 720*x^11 - 111*x^10 - 4259*x^9 + 1809*x^8 + 10989*x^7 - 7125*x^6 - 11826*x^5 + 10260*x^4 + 3483*x^3 - 4617*x^2 + 486*x + 243, 1)
 

Normalized defining polynomial

\( x^{15} - 48 x^{13} - 4 x^{12} + 720 x^{11} - 111 x^{10} - 4259 x^{9} + 1809 x^{8} + 10989 x^{7} - 7125 x^{6} - 11826 x^{5} + 10260 x^{4} + 3483 x^{3} - 4617 x^{2} + 486 x + 243 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(401095849317140421258328161=3^{19}\cdot 83\cdot 401^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.37$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 83, 401$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{18} a^{10} - \frac{1}{18} a^{9} - \frac{1}{9} a^{8} + \frac{1}{18} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{18} a^{11} - \frac{1}{6} a^{9} - \frac{1}{18} a^{8} - \frac{1}{6} a^{7} - \frac{4}{9} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{54} a^{12} + \frac{1}{27} a^{9} + \frac{1}{18} a^{8} - \frac{1}{6} a^{7} + \frac{2}{27} a^{6} + \frac{2}{9} a^{5} - \frac{1}{3} a^{4} - \frac{5}{18} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{702} a^{13} + \frac{1}{702} a^{12} + \frac{2}{117} a^{11} + \frac{1}{351} a^{10} - \frac{1}{54} a^{9} + \frac{4}{39} a^{8} + \frac{103}{702} a^{7} - \frac{1}{351} a^{6} - \frac{29}{117} a^{5} - \frac{11}{234} a^{4} + \frac{61}{234} a^{3} + \frac{4}{39} a^{2} - \frac{1}{2} a - \frac{5}{26}$, $\frac{1}{12161937294} a^{14} + \frac{924880}{2026989549} a^{13} - \frac{6075161}{2026989549} a^{12} + \frac{102364207}{6080968647} a^{11} - \frac{24353783}{4053979098} a^{10} - \frac{157556557}{4053979098} a^{9} + \frac{750953159}{6080968647} a^{8} + \frac{196972237}{2026989549} a^{7} + \frac{20030773}{225221061} a^{6} + \frac{1964409271}{4053979098} a^{5} - \frac{293415302}{675663183} a^{4} - \frac{99030775}{225221061} a^{3} + \frac{130240205}{450442122} a^{2} + \frac{35960259}{75073687} a + \frac{28501449}{75073687}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 443934700.445 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T86:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 77760
The 72 conjugacy class representatives for [S(3)^5]D(5)=S(3)wrD(5) are not computed
Character table for [S(3)^5]D(5)=S(3)wrD(5) is not computed

Intermediate fields

5.5.160801.1

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
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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ R $15$ $15$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$3.3.5.2$x^{3} + 21$$3$$1$$5$$S_3$$[5/2]_{2}$
3.6.6.2$x^{6} + 6 x^{4} + 6 x^{3} + 18$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
3.6.8.1$x^{6} + 6 x^{5} + 18 x^{2} + 9$$3$$2$$8$$C_3^2:C_4$$[2, 2]^{4}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
83.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
401Data not computed