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Properties

Label 15.15.4167625760...7952.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 3^{20}\cdot 7^{4}\cdot 36497^{3}$
Root discriminant $94.33$
Ramified primes $2, 3, 7, 36497$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T91

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3146976, -9751392, 4127760, 9263884, -1845144, -3388707, 391884, 626151, -42696, -63251, 2268, 3483, -46, -96, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 96*x^13 - 46*x^12 + 3483*x^11 + 2268*x^10 - 63251*x^9 - 42696*x^8 + 626151*x^7 + 391884*x^6 - 3388707*x^5 - 1845144*x^4 + 9263884*x^3 + 4127760*x^2 - 9751392*x - 3146976)
 
gp: K = bnfinit(x^15 - 96*x^13 - 46*x^12 + 3483*x^11 + 2268*x^10 - 63251*x^9 - 42696*x^8 + 626151*x^7 + 391884*x^6 - 3388707*x^5 - 1845144*x^4 + 9263884*x^3 + 4127760*x^2 - 9751392*x - 3146976, 1)
 

Normalized defining polynomial

\( x^{15} - 96 x^{13} - 46 x^{12} + 3483 x^{11} + 2268 x^{10} - 63251 x^{9} - 42696 x^{8} + 626151 x^{7} + 391884 x^{6} - 3388707 x^{5} - 1845144 x^{4} + 9263884 x^{3} + 4127760 x^{2} - 9751392 x - 3146976 \)

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:  \(416762576016182015933206397952=2^{10}\cdot 3^{20}\cdot 7^{4}\cdot 36497^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.33$
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:  $2, 3, 7, 36497$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{28} a^{12} - \frac{3}{7} a^{10} + \frac{5}{14} a^{9} + \frac{11}{28} a^{8} + \frac{1}{28} a^{6} + \frac{1}{7} a^{5} - \frac{13}{28} a^{4} - \frac{1}{7} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{56} a^{13} - \frac{3}{14} a^{11} + \frac{5}{28} a^{10} + \frac{11}{56} a^{9} - \frac{1}{2} a^{8} + \frac{1}{56} a^{7} - \frac{3}{7} a^{6} - \frac{13}{56} a^{5} - \frac{1}{14} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{131826257606797798840303467161328} a^{14} - \frac{22930472973966603390742021805}{5492760733616574951679311131722} a^{13} - \frac{16884024519853141928472137275}{5492760733616574951679311131722} a^{12} + \frac{11523872515340701164494383441117}{65913128803398899420151733580664} a^{11} - \frac{9922521280418381935156923988903}{43942085868932599613434489053776} a^{10} + \frac{282071213232469493106785511527}{10985521467233149903358622263444} a^{9} - \frac{50596787232216004429608752347931}{131826257606797798840303467161328} a^{8} - \frac{2008745955639789054649661600955}{5492760733616574951679311131722} a^{7} - \frac{6973964970859449076644559749531}{43942085868932599613434489053776} a^{6} - \frac{861421373522514945509829378059}{10985521467233149903358622263444} a^{5} - \frac{18789972886977346582074312519209}{43942085868932599613434489053776} a^{4} + \frac{28341560932955430695717817673}{784680104802367850239901590246} a^{3} + \frac{33968558540563528259712939499}{4708080628814207101439409541476} a^{2} + \frac{203159431568676633119091010749}{784680104802367850239901590246} a - \frac{20791556922555783314066830970}{392340052401183925119950795123}$

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:  \( 11897255378.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T91:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 466560
The 72 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

5.5.36497.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$3$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.12.16.18$x^{12} - 93 x^{11} + 9 x^{10} - 93 x^{9} - 27 x^{8} + 81 x^{7} + 108 x^{6} - 81 x^{5} - 81 x^{4} - 81$$3$$4$$16$12T173$[2, 2, 2, 2]^{8}$
$7$7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.9.0.1$x^{9} + x^{2} - 6 x + 2$$1$$9$$0$$C_9$$[\ ]^{9}$
36497Data not computed