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Properties

Label 15.5.12170194953...0000.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{18}\cdot 5^{6}\cdot 7^{4}\cdot 23^{7}\cdot 2819^{4}\cdot 2399027^{2}$
Root discriminant $1872.20$
Ramified primes $2, 5, 7, 23, 2819, 2399027$
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![11546624000, 0, -14577612800, -11221875200, 4806282240, 7254954240, 2578282496, -230315360, -113494464, -16875000, 645184, 196268, -288, -792, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 792*x^13 - 288*x^12 + 196268*x^11 + 645184*x^10 - 16875000*x^9 - 113494464*x^8 - 230315360*x^7 + 2578282496*x^6 + 7254954240*x^5 + 4806282240*x^4 - 11221875200*x^3 - 14577612800*x^2 + 11546624000)
 
gp: K = bnfinit(x^15 - 792*x^13 - 288*x^12 + 196268*x^11 + 645184*x^10 - 16875000*x^9 - 113494464*x^8 - 230315360*x^7 + 2578282496*x^6 + 7254954240*x^5 + 4806282240*x^4 - 11221875200*x^3 - 14577612800*x^2 + 11546624000, 1)
 

Normalized defining polynomial

\( x^{15} - 792 x^{13} - 288 x^{12} + 196268 x^{11} + 645184 x^{10} - 16875000 x^{9} - 113494464 x^{8} - 230315360 x^{7} + 2578282496 x^{6} + 7254954240 x^{5} + 4806282240 x^{4} - 11221875200 x^{3} - 14577612800 x^{2} + 11546624000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-12170194953371312791319056980973540985335808000000=-\,2^{18}\cdot 5^{6}\cdot 7^{4}\cdot 23^{7}\cdot 2819^{4}\cdot 2399027^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1872.20$
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:  $2, 5, 7, 23, 2819, 2399027$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{16} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{8} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{9} - \frac{1}{16} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{640} a^{10} - \frac{1}{160} a^{9} + \frac{1}{160} a^{8} + \frac{1}{40} a^{7} - \frac{9}{160} a^{6} + \frac{3}{40} a^{5} - \frac{9}{80} a^{4} - \frac{3}{20} a^{3} + \frac{1}{10} a^{2}$, $\frac{1}{1280} a^{11} + \frac{1}{160} a^{9} - \frac{1}{160} a^{8} + \frac{7}{320} a^{7} + \frac{1}{20} a^{6} - \frac{3}{32} a^{5} + \frac{3}{40} a^{4} - \frac{1}{8} a^{3} - \frac{1}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{2560} a^{12} - \frac{1}{160} a^{9} - \frac{1}{640} a^{8} - \frac{1}{40} a^{7} - \frac{19}{320} a^{6} - \frac{1}{20} a^{5} - \frac{7}{80} a^{4} + \frac{3}{20} a^{3} + \frac{1}{20} a^{2}$, $\frac{1}{51200} a^{13} + \frac{1}{6400} a^{11} + \frac{1}{1600} a^{10} - \frac{53}{12800} a^{9} + \frac{1}{800} a^{8} + \frac{13}{1280} a^{7} - \frac{31}{800} a^{6} + \frac{5}{64} a^{5} - \frac{3}{25} a^{4} - \frac{1}{40} a^{3} + \frac{1}{10} a^{2} + \frac{1}{4} a$, $\frac{1}{9174740086938070601701826763947174856436488505360998400} a^{14} - \frac{2747648275118399658556299614881546900013357681841}{286710627716814706303182086373349214263640265792531200} a^{13} + \frac{117978099213882676903953270266381529289406996753761}{1146842510867258825212728345493396857054561063170124800} a^{12} + \frac{29333294073587537098552727168946608557106194695469}{143355313858407353151591043186674607131820132896265600} a^{11} - \frac{1765680405550315385524652109554510680964737108883749}{2293685021734517650425456690986793714109122126340249600} a^{10} - \frac{531259894959059395280795504795470825538450831393113}{143355313858407353151591043186674607131820132896265600} a^{9} - \frac{54132891373629680327388117247129873953732693342351}{1146842510867258825212728345493396857054561063170124800} a^{8} - \frac{1123013535170949464154099605218370756359823307454561}{143355313858407353151591043186674607131820132896265600} a^{7} - \frac{572170334979306409903975781960324076458237286042651}{286710627716814706303182086373349214263640265792531200} a^{6} - \frac{2184184157242016525650338015491649826015094221593089}{35838828464601838287897760796668651782955033224066400} a^{5} - \frac{3797222879386076135570034771049760113153270254413757}{35838828464601838287897760796668651782955033224066400} a^{4} - \frac{15095710439405104411256275162967326756037597738995}{89597071161504595719744401991671629457387583060166} a^{3} + \frac{769145065973238030761437367277855938641363522180123}{3583882846460183828789776079666865178295503322406640} a^{2} + \frac{83567593991873933750934640651183271391529405209359}{179194142323009191439488803983343258914775166120332} a - \frac{17865558091208057342142013938269270850523527039557}{44798535580752297859872200995835814728693791530083}$

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:  $9$
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:  \( 6577313709220000000 \) (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

3.1.23.1

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 $15$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $15$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R $15$ $15$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.2$x^{6} + 4 x^{2} - 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.9.2$x^{6} + 4 x^{2} - 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$5$$\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$$[\ ]_{2}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.10.5.2$x^{10} - 279841 x^{2} + 12872686$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
2819Data not computed
2399027Data not computed