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Properties

Label 15.15.6117089244...6561.1
Degree $15$
Signature $[15, 0]$
Discriminant $193^{2}\cdot 89417^{4}\cdot 5068447^{2}$
Root discriminant $330.45$
Ramified primes $193, 89417, 5068447$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T89

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-21039807, -107138692, 30695428, 105247589, -10209282, -35856634, 731596, 5468724, 100840, -391193, -13709, 12766, 438, -186, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 - 186*x^13 + 438*x^12 + 12766*x^11 - 13709*x^10 - 391193*x^9 + 100840*x^8 + 5468724*x^7 + 731596*x^6 - 35856634*x^5 - 10209282*x^4 + 105247589*x^3 + 30695428*x^2 - 107138692*x - 21039807)
 
gp: K = bnfinit(x^15 - 4*x^14 - 186*x^13 + 438*x^12 + 12766*x^11 - 13709*x^10 - 391193*x^9 + 100840*x^8 + 5468724*x^7 + 731596*x^6 - 35856634*x^5 - 10209282*x^4 + 105247589*x^3 + 30695428*x^2 - 107138692*x - 21039807, 1)
 

Normalized defining polynomial

\( x^{15} - 4 x^{14} - 186 x^{13} + 438 x^{12} + 12766 x^{11} - 13709 x^{10} - 391193 x^{9} + 100840 x^{8} + 5468724 x^{7} + 731596 x^{6} - 35856634 x^{5} - 10209282 x^{4} + 105247589 x^{3} + 30695428 x^{2} - 107138692 x - 21039807 \)

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:  \(61170892447385797247321113311726286561=193^{2}\cdot 89417^{4}\cdot 5068447^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $330.45$
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:  $193, 89417, 5068447$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{25} a^{13} - \frac{1}{25} a^{12} - \frac{2}{5} a^{11} + \frac{4}{25} a^{10} - \frac{12}{25} a^{9} - \frac{4}{25} a^{8} - \frac{3}{25} a^{7} - \frac{2}{5} a^{6} + \frac{7}{25} a^{5} + \frac{2}{25} a^{4} + \frac{1}{25} a^{2} - \frac{8}{25} a + \frac{8}{25}$, $\frac{1}{1082103447742775892067492242798723356217914122625} a^{14} + \frac{593474016545591530432603846031886245970857728}{43284137909711035682699689711948934248716564905} a^{13} - \frac{27448054592937759819450953444193729099939557186}{1082103447742775892067492242798723356217914122625} a^{12} + \frac{370584858866103009357742320775404112867842649669}{1082103447742775892067492242798723356217914122625} a^{11} + \frac{314488602733335190652154846087054673414888479992}{1082103447742775892067492242798723356217914122625} a^{10} - \frac{406519631908030904982896364310311027222081015791}{1082103447742775892067492242798723356217914122625} a^{9} + \frac{2869014670223759298047855226985088169022532293}{1082103447742775892067492242798723356217914122625} a^{8} + \frac{276229520039809659975229606820042296503524651312}{1082103447742775892067492242798723356217914122625} a^{7} + \frac{61285324272216242216702875852172752150526400772}{1082103447742775892067492242798723356217914122625} a^{6} + \frac{143384104955984482692042235806919237782040104209}{1082103447742775892067492242798723356217914122625} a^{5} + \frac{104312204466393929760061310394054617515860306652}{1082103447742775892067492242798723356217914122625} a^{4} - \frac{306346529310069442823471127459619648430186851999}{1082103447742775892067492242798723356217914122625} a^{3} - \frac{204724050272881626148898580094075756715851444957}{1082103447742775892067492242798723356217914122625} a^{2} + \frac{7252509952762224421044306080379529742390209856}{43284137909711035682699689711948934248716564905} a - \frac{272960364578721969243685912407462125773372871317}{1082103447742775892067492242798723356217914122625}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 121855598236000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T89:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 466560
The 60 conjugacy class representatives for 1/2[S(3)^5]S(5) are not computed
Character table for 1/2[S(3)^5]S(5) is not computed

Intermediate fields

5.5.89417.1

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 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 $15$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ $15$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
193Data not computed
89417Data not computed
5068447Data not computed