↑

Properties

Label 15.11.1543040910...5041.1
Degree $15$
Signature $[11, 2]$
Discriminant $61^{4}\cdot 397^{4}\cdot 28309^{2}\cdot 74821^{2}$
Root discriminant $258.56$
Ramified primes $61, 397, 28309, 74821$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T89

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![120987661, -423492779, 540249339, -288701374, 26547802, 37048867, -13231140, -622189, 1029924, -95880, -34514, 5769, 531, -126, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 126*x^13 + 531*x^12 + 5769*x^11 - 34514*x^10 - 95880*x^9 + 1029924*x^8 - 622189*x^7 - 13231140*x^6 + 37048867*x^5 + 26547802*x^4 - 288701374*x^3 + 540249339*x^2 - 423492779*x + 120987661)
 
gp: K = bnfinit(x^15 - 3*x^14 - 126*x^13 + 531*x^12 + 5769*x^11 - 34514*x^10 - 95880*x^9 + 1029924*x^8 - 622189*x^7 - 13231140*x^6 + 37048867*x^5 + 26547802*x^4 - 288701374*x^3 + 540249339*x^2 - 423492779*x + 120987661, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 126 x^{13} + 531 x^{12} + 5769 x^{11} - 34514 x^{10} - 95880 x^{9} + 1029924 x^{8} - 622189 x^{7} - 13231140 x^{6} + 37048867 x^{5} + 26547802 x^{4} - 288701374 x^{3} + 540249339 x^{2} - 423492779 x + 120987661 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1543040910519569546921019055910525041=61^{4}\cdot 397^{4}\cdot 28309^{2}\cdot 74821^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $258.56$
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:  $61, 397, 28309, 74821$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{18839329822924913935029637071464706293657} a^{14} + \frac{1559883742475307643250532698550376482198}{18839329822924913935029637071464706293657} a^{13} - \frac{6930805050881772307642496663303894317773}{18839329822924913935029637071464706293657} a^{12} + \frac{4277067551368383413588733366680038519882}{18839329822924913935029637071464706293657} a^{11} + \frac{7680304512736244984331700698020368937162}{18839329822924913935029637071464706293657} a^{10} - \frac{2992987637870185535072949319117001489362}{18839329822924913935029637071464706293657} a^{9} + \frac{8848799924595544649020284199316074653690}{18839329822924913935029637071464706293657} a^{8} - \frac{6032048163941785368945569859410770190050}{18839329822924913935029637071464706293657} a^{7} - \frac{6680109500740416304457424236347165844258}{18839329822924913935029637071464706293657} a^{6} - \frac{5401124151457369788189445155425408158513}{18839329822924913935029637071464706293657} a^{5} + \frac{5389006382407330417669885416025807935367}{18839329822924913935029637071464706293657} a^{4} + \frac{8600879111930944366723636493539044371463}{18839329822924913935029637071464706293657} a^{3} - \frac{5962029872383427794596609585760586303919}{18839329822924913935029637071464706293657} a^{2} + \frac{264850536937734587614852489927328226450}{18839329822924913935029637071464706293657} a - \frac{6537566668048194912229484184795389966025}{18839329822924913935029637071464706293657}$

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:  $12$
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:  \( 1522175959660 \) (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

5.5.24217.1

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 $15$ $15$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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
61Data not computed
397Data not computed
28309Data not computed
74821Data not computed