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Properties

Label 15.11.3718506120...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{21}\cdot 5^{6}\cdot 13\cdot 37^{5}\cdot 293^{2}\cdot 2549^{4}\cdot 18637^{2}$
Root discriminant $1272.60$
Ramified primes $2, 5, 13, 37, 293, 2549, 18637$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T102

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-326272000, 2283904000, -5522153600, 4945060000, -146822400, -1799651680, 426992368, 189147496, -46796256, -5022270, 1097648, 77582, -7416, -720, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 720*x^13 - 7416*x^12 + 77582*x^11 + 1097648*x^10 - 5022270*x^9 - 46796256*x^8 + 189147496*x^7 + 426992368*x^6 - 1799651680*x^5 - 146822400*x^4 + 4945060000*x^3 - 5522153600*x^2 + 2283904000*x - 326272000)
 
gp: K = bnfinit(x^15 - 720*x^13 - 7416*x^12 + 77582*x^11 + 1097648*x^10 - 5022270*x^9 - 46796256*x^8 + 189147496*x^7 + 426992368*x^6 - 1799651680*x^5 - 146822400*x^4 + 4945060000*x^3 - 5522153600*x^2 + 2283904000*x - 326272000, 1)
 

Normalized defining polynomial

\( x^{15} - 720 x^{13} - 7416 x^{12} + 77582 x^{11} + 1097648 x^{10} - 5022270 x^{9} - 46796256 x^{8} + 189147496 x^{7} + 426992368 x^{6} - 1799651680 x^{5} - 146822400 x^{4} + 4945060000 x^{3} - 5522153600 x^{2} + 2283904000 x - 326272000 \)

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:  \(37185061201306863596258469574643558678528000000=2^{21}\cdot 5^{6}\cdot 13\cdot 37^{5}\cdot 293^{2}\cdot 2549^{4}\cdot 18637^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1272.60$
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:  $2, 5, 13, 37, 293, 2549, 18637$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{16} a^{10} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{160} a^{11} + \frac{1}{40} a^{8} + \frac{11}{80} a^{7} - \frac{1}{5} a^{6} + \frac{5}{16} a^{5} - \frac{7}{20} a^{4} + \frac{7}{20} a^{3} + \frac{1}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{320} a^{12} + \frac{1}{80} a^{9} + \frac{11}{160} a^{8} + \frac{3}{20} a^{7} + \frac{5}{32} a^{6} + \frac{13}{40} a^{5} + \frac{7}{40} a^{4} - \frac{19}{40} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1600} a^{13} - \frac{1}{100} a^{10} - \frac{9}{800} a^{9} + \frac{2}{25} a^{8} - \frac{3}{160} a^{7} + \frac{19}{100} a^{6} - \frac{63}{200} a^{5} - \frac{11}{50} a^{4} - \frac{1}{10} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{5860948555055282394966333512611156453507952637446816000} a^{14} + \frac{21438049090922799566558256502373282392105538822241}{73261856938191029937079168907639455668849407968085200} a^{13} - \frac{2384637307602424169463220015504953331567707299879}{1831546423454775748426979222690986391721235199202130} a^{12} - \frac{2155396188735113888038833099195210237063123786821027}{732618569381910299370791689076394556688494079680852000} a^{11} - \frac{76560371460294433738772319285624336273891330191089449}{2930474277527641197483166756305578226753976318723408000} a^{10} + \frac{3295364805455511519119549560742999381484807845386743}{366309284690955149685395844538197278344247039840426000} a^{9} - \frac{45256042261035178809258045360951757906716904895235619}{586094855505528239496633351261115645350795263744681600} a^{8} + \frac{3223373479395734402974849127959858327778091566020959}{22894330293184696855337240283637329896515439990026625} a^{7} + \frac{126758829693342743380223105040913654569656330752597577}{732618569381910299370791689076394556688494079680852000} a^{6} - \frac{71329067690015817293206776973463599833106293293227817}{366309284690955149685395844538197278344247039840426000} a^{5} + \frac{13918907454405525861987295973872460530843027246135743}{36630928469095514968539584453819727834424703984042600} a^{4} + \frac{2267374096768589400004850557754763585971531826480463}{18315464234547757484269792226909863917212351992021300} a^{3} + \frac{439835445129060284671364913223570116195018111802749}{7326185693819102993707916890763945566884940796808520} a^{2} - \frac{20113966017582696560616313789347874218023712629247}{1831546423454775748426979222690986391721235199202130} a - \frac{91243489741114916276886617613652190209433913981393}{183154642345477574842697922269098639172123519920213}$

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:  $12$
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:  \( 16057293665000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T102:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10368000
The 140 conjugacy class representatives for [S(5)^3]S(3)=S(5)wrS(3) are not computed
Character table for [S(5)^3]S(3)=S(5)wrS(3) is not computed

Intermediate fields

3.3.148.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ R ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.11.15$x^{6} + 2 x^{4} + 2 x^{2} + 10$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.10.0.1$x^{10} + 2 x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$37$37.5.0.1$x^{5} - x + 13$$1$$5$$0$$C_5$$[\ ]^{5}$
37.10.5.1$x^{10} - 2738 x^{6} + 1874161 x^{2} - 11719128733$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
293Data not computed
2549Data not computed
18637Data not computed