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Properties

Label 15.7.87173191285...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{24}\cdot 5^{6}\cdot 13^{2}\cdot 29^{4}\cdot 37^{5}\cdot 151^{2}\cdot 41947^{2}$
Root discriminant $536.24$
Ramified primes $2, 5, 13, 29, 37, 151, 41947$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T96

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-464000, -3248000, -6681600, -1537000, 6973920, 4312160, -1669256, -1412708, 94320, 182394, 37432, 754, -1008, -144, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 144*x^13 - 1008*x^12 + 754*x^11 + 37432*x^10 + 182394*x^9 + 94320*x^8 - 1412708*x^7 - 1669256*x^6 + 4312160*x^5 + 6973920*x^4 - 1537000*x^3 - 6681600*x^2 - 3248000*x - 464000)
 
gp: K = bnfinit(x^15 - 144*x^13 - 1008*x^12 + 754*x^11 + 37432*x^10 + 182394*x^9 + 94320*x^8 - 1412708*x^7 - 1669256*x^6 + 4312160*x^5 + 6973920*x^4 - 1537000*x^3 - 6681600*x^2 - 3248000*x - 464000, 1)
 

Normalized defining polynomial

\( x^{15} - 144 x^{13} - 1008 x^{12} + 754 x^{11} + 37432 x^{10} + 182394 x^{9} + 94320 x^{8} - 1412708 x^{7} - 1669256 x^{6} + 4312160 x^{5} + 6973920 x^{4} - 1537000 x^{3} - 6681600 x^{2} - 3248000 x - 464000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(87173191285397298520836997842731008000000=2^{24}\cdot 5^{6}\cdot 13^{2}\cdot 29^{4}\cdot 37^{5}\cdot 151^{2}\cdot 41947^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $536.24$
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:  $2, 5, 13, 29, 37, 151, 41947$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{20} a^{11} + \frac{1}{20} a^{9} + \frac{1}{10} a^{8} + \frac{1}{5} a^{7} + \frac{1}{10} a^{6} + \frac{1}{5} a^{5} + \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{120} a^{12} - \frac{1}{60} a^{11} + \frac{1}{20} a^{10} + \frac{1}{12} a^{9} + \frac{1}{12} a^{8} + \frac{1}{30} a^{7} - \frac{1}{12} a^{6} - \frac{2}{5} a^{5} - \frac{1}{15} a^{4} - \frac{1}{2} a^{3} - \frac{1}{15} a^{2} + \frac{1}{3}$, $\frac{1}{1200} a^{13} - \frac{1}{300} a^{11} - \frac{1}{150} a^{10} - \frac{1}{200} a^{9} - \frac{6}{25} a^{8} + \frac{77}{600} a^{7} - \frac{1}{6} a^{6} + \frac{13}{300} a^{5} + \frac{43}{150} a^{4} - \frac{7}{15} a^{3} + \frac{1}{15} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{64883499766640909367514567989779258400} a^{14} - \frac{6564230388366609821440973402924309}{16220874941660227341878641997444814600} a^{13} + \frac{5914888542152833498644769813310371}{4055218735415056835469660499361203650} a^{12} + \frac{187565132110337923893149543853568297}{8110437470830113670939320998722407300} a^{11} + \frac{609330061754669510388297391794190661}{32441749883320454683757283994889629200} a^{10} - \frac{604023497335707323605754816227838969}{8110437470830113670939320998722407300} a^{9} - \frac{1227343641905109087081937953786008073}{10813916627773484894585761331629876400} a^{8} - \frac{29433164594360563127005543367897113}{8110437470830113670939320998722407300} a^{7} + \frac{1066757127294250349783955481157009541}{5406958313886742447292880665814938200} a^{6} - \frac{1198382041070286606863352817700779957}{2703479156943371223646440332907469100} a^{5} + \frac{179737818289919233487049508169573021}{4055218735415056835469660499361203650} a^{4} + \frac{31828637980212193477008881795159217}{270347915694337122364644033290746910} a^{3} - \frac{593534122778493308483853900066354673}{1622087494166022734187864199744481460} a^{2} - \frac{7377732803630853881505131681323492}{27034791569433712236464403329074691} a + \frac{24601217958519280765743297552771164}{81104374708301136709393209987224073}$

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:  $10$
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:  \( 2898519216390000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T96:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1296000
The 65 conjugacy class representatives for [A(5)^3]S(3)=A(5)wrS(3) are not computed
Character table for [A(5)^3]S(3)=A(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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ R ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ R $15$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$[\ ]_{3}^{2}$
2.6.11.8$x^{6} + 4 x^{4} + 2 x^{2} + 6$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
2.6.11.8$x^{6} + 4 x^{4} + 2 x^{2} + 6$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
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}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$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$Trivial$[\ ]$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.10.0.1$x^{10} + 2 x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$29$29.5.4.1$x^{5} - 29$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
37Data not computed
151Data not computed
41947Data not computed