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Properties

Label 15.1.88308891684...0944.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{24}\cdot 31^{5}\cdot 3691^{2}\cdot 11617^{2}$
Root discriminant $99.17$
Ramified primes $2, 31, 3691, 11617$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T74

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-26757, -78572, -30041, -10819, 9134, 158780, -133204, 36398, -13223, 7210, -1439, 257, -135, 18, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + 18*x^13 - 135*x^12 + 257*x^11 - 1439*x^10 + 7210*x^9 - 13223*x^8 + 36398*x^7 - 133204*x^6 + 158780*x^5 + 9134*x^4 - 10819*x^3 - 30041*x^2 - 78572*x - 26757)
 
gp: K = bnfinit(x^15 - x^14 + 18*x^13 - 135*x^12 + 257*x^11 - 1439*x^10 + 7210*x^9 - 13223*x^8 + 36398*x^7 - 133204*x^6 + 158780*x^5 + 9134*x^4 - 10819*x^3 - 30041*x^2 - 78572*x - 26757, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} + 18 x^{13} - 135 x^{12} + 257 x^{11} - 1439 x^{10} + 7210 x^{9} - 13223 x^{8} + 36398 x^{7} - 133204 x^{6} + 158780 x^{5} + 9134 x^{4} - 10819 x^{3} - 30041 x^{2} - 78572 x - 26757 \)

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:  \(-883088916844315174708071890944=-\,2^{24}\cdot 31^{5}\cdot 3691^{2}\cdot 11617^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.17$
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:  $2, 31, 3691, 11617$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{3}{8} a^{2} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{11} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{8} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{1888284870356225904031001940136} a^{14} + \frac{25929588062474163360073491083}{472071217589056476007750485034} a^{13} - \frac{6371353495848523970043048507}{472071217589056476007750485034} a^{12} + \frac{74390707882289151530547007647}{1888284870356225904031001940136} a^{11} - \frac{38693177653731191314227852765}{1888284870356225904031001940136} a^{10} - \frac{4005589455063277720115245703}{236035608794528238003875242517} a^{9} - \frac{198944319731711683632167604769}{1888284870356225904031001940136} a^{8} + \frac{57267806237434358475723068694}{236035608794528238003875242517} a^{7} + \frac{456535191832274532794523889809}{1888284870356225904031001940136} a^{6} + \frac{113836010439410725727756659900}{236035608794528238003875242517} a^{5} + \frac{51905678359121151808505224309}{472071217589056476007750485034} a^{4} + \frac{532505667065209424225196143307}{1888284870356225904031001940136} a^{3} - \frac{79173166816102818439335430628}{236035608794528238003875242517} a^{2} - \frac{723740891191832250398996948791}{1888284870356225904031001940136} a + \frac{95456135937838312177103024630}{236035608794528238003875242517}$

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:  $7$
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:  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:  \( 1125442805.24 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T74:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24000
The 40 conjugacy class representatives for [1/2.F(5)^3]S(3)
Character table for [1/2.F(5)^3]S(3) is not computed

Intermediate fields

3.1.31.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ R ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.24.231$x^{12} + 24 x^{11} - 26 x^{10} - 20 x^{9} + 6 x^{8} + 24 x^{7} + 16 x^{6} - 16 x^{5} - 4 x^{4} - 16 x^{3} - 8 x^{2} + 16 x - 24$$4$$3$$24$12T55$[2, 2, 3, 3, 3]^{3}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.8.4.1$x^{8} + 32674 x^{4} - 119164 x^{2} + 266897569$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3691Data not computed
11617Data not computed