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Properties

Label 15.5.21897430124...1792.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{10}\cdot 3^{5}\cdot 11^{5}\cdot 31^{13}\cdot 181^{13}$
Root discriminant $9037.04$
Ramified primes $2, 3, 11, 31, 181$
Class number Not computed
Class group Not computed
Galois group $S_3 \times C_5$ (as 15T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-156585180055842346293, 20744208523450960470, 5950390757960217555, -1205008059400695846, 131630106918356613, -10215903964721319, 579004818046875, -23558245878420, 531822012342, 3771529909, -586917623, 10337726, 145177, -5518, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 5518*x^13 + 145177*x^12 + 10337726*x^11 - 586917623*x^10 + 3771529909*x^9 + 531822012342*x^8 - 23558245878420*x^7 + 579004818046875*x^6 - 10215903964721319*x^5 + 131630106918356613*x^4 - 1205008059400695846*x^3 + 5950390757960217555*x^2 + 20744208523450960470*x - 156585180055842346293)
 
gp: K = bnfinit(x^15 - 2*x^14 - 5518*x^13 + 145177*x^12 + 10337726*x^11 - 586917623*x^10 + 3771529909*x^9 + 531822012342*x^8 - 23558245878420*x^7 + 579004818046875*x^6 - 10215903964721319*x^5 + 131630106918356613*x^4 - 1205008059400695846*x^3 + 5950390757960217555*x^2 + 20744208523450960470*x - 156585180055842346293, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 5518 x^{13} + 145177 x^{12} + 10337726 x^{11} - 586917623 x^{10} + 3771529909 x^{9} + 531822012342 x^{8} - 23558245878420 x^{7} + 579004818046875 x^{6} - 10215903964721319 x^{5} + 131630106918356613 x^{4} - 1205008059400695846 x^{3} + 5950390757960217555 x^{2} + 20744208523450960470 x - 156585180055842346293 \)

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:  \(-218974301248546884855355633871639977128201417943833178401792=-\,2^{10}\cdot 3^{5}\cdot 11^{5}\cdot 31^{13}\cdot 181^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $9037.04$
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:  $2, 3, 11, 31, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{6} - \frac{2}{27} a^{4} - \frac{1}{9} a^{3} + \frac{1}{27} a^{2} + \frac{1}{9} a$, $\frac{1}{27} a^{7} + \frac{1}{27} a^{5} + \frac{1}{9} a^{4} - \frac{2}{27} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{8} - \frac{1}{81} a^{7} + \frac{1}{81} a^{6} - \frac{4}{81} a^{5} + \frac{1}{81} a^{4} - \frac{4}{81} a^{3} + \frac{2}{27} a^{2}$, $\frac{1}{243} a^{9} - \frac{1}{243} a^{8} + \frac{4}{243} a^{7} - \frac{4}{243} a^{6} + \frac{13}{243} a^{5} + \frac{23}{243} a^{4} - \frac{1}{27} a^{3} - \frac{1}{3} a^{2} + \frac{2}{9} a$, $\frac{1}{4090419} a^{10} + \frac{5854}{4090419} a^{9} + \frac{18785}{4090419} a^{8} - \frac{73157}{4090419} a^{7} + \frac{19040}{4090419} a^{6} - \frac{79115}{4090419} a^{5} + \frac{144298}{4090419} a^{4} - \frac{223711}{1363473} a^{3} + \frac{86011}{454491} a^{2} + \frac{6379}{50499} a - \frac{60}{5611}$, $\frac{1}{36813771} a^{11} - \frac{2}{36813771} a^{10} - \frac{7084}{36813771} a^{9} - \frac{5117}{1187541} a^{8} + \frac{110747}{36813771} a^{7} - \frac{193394}{36813771} a^{6} - \frac{1384724}{36813771} a^{5} + \frac{1568702}{12271257} a^{4} - \frac{293351}{4090419} a^{3} + \frac{43409}{454491} a^{2} - \frac{11615}{50499} a - \frac{237}{5611}$, $\frac{1}{35009896221} a^{12} + \frac{214}{35009896221} a^{11} - \frac{3196}{35009896221} a^{10} + \frac{38144221}{35009896221} a^{9} - \frac{196608661}{35009896221} a^{8} + \frac{495322621}{35009896221} a^{7} + \frac{414504538}{35009896221} a^{6} - \frac{470359411}{11669965407} a^{5} - \frac{119512298}{3889988469} a^{4} - \frac{60145598}{432220941} a^{3} + \frac{27591817}{144073647} a^{2} + \frac{1821200}{5336061} a + \frac{511992}{1778687}$, $\frac{1}{945267197967} a^{13} - \frac{8}{945267197967} a^{12} - \frac{7909}{945267197967} a^{11} + \frac{72904}{945267197967} a^{10} + \frac{1511434706}{945267197967} a^{9} + \frac{2737812118}{945267197967} a^{8} - \frac{9362361878}{945267197967} a^{7} - \frac{11131174}{105029688663} a^{6} + \frac{1078926965}{35009896221} a^{5} + \frac{4183238080}{35009896221} a^{4} + \frac{301650466}{3889988469} a^{3} - \frac{64897718}{432220941} a^{2} - \frac{236330}{1778687} a - \frac{557238}{1778687}$, $\frac{1}{294933351974471864399275982800812245804938184001302803636160582256756864742513515447} a^{14} - \frac{44630463166023064535463724526981695798703742108011847867852363470522441}{294933351974471864399275982800812245804938184001302803636160582256756864742513515447} a^{13} - \frac{2699672476110508279970965049290973465763063436200251470869403925515463698}{294933351974471864399275982800812245804938184001302803636160582256756864742513515447} a^{12} + \frac{747236486393927245586109787442876592504938365411605548923885407733100196460}{294933351974471864399275982800812245804938184001302803636160582256756864742513515447} a^{11} - \frac{7941687075348935631384055545498466953000405974289787768983548959150546019467}{294933351974471864399275982800812245804938184001302803636160582256756864742513515447} a^{10} + \frac{16533032371323088013766595910351691899759473771373922322183407336995333250559858}{9513979095950705303202451058090717606610909161332348504392276846992156927177855337} a^{9} + \frac{987591017625183540758506698635110735472405339915612382330485690604259005191941219}{294933351974471864399275982800812245804938184001302803636160582256756864742513515447} a^{8} + \frac{201657775282317310436684961637398846752141653106067103699176137145494292761179105}{98311117324823954799758660933604081934979394667100934545386860752252288247504505149} a^{7} - \frac{110137117781049317811338558830047895125458986617808908757630751894788929636779888}{10923457480535994977750962325956009103886599407455659393931873416916920916389389461} a^{6} + \frac{152778380954473170974041016183012230887925515093163139710721058915908480758677304}{10923457480535994977750962325956009103886599407455659393931873416916920916389389461} a^{5} + \frac{471321188744283651169729581782480804549965785078274642072126962382615794605645179}{3641152493511998325916987441985336367962199802485219797977291138972306972129796487} a^{4} - \frac{30666518142690801953365864117220646972992973802507415097646229973660874830524513}{404572499279110925101887493553926263106911089165024421997476793219145219125532943} a^{3} + \frac{14612880341160749758438619470870301162929951650672343661011913020077694452508666}{44952499919901213900209721505991807011879009907224935777497421468793913236170327} a^{2} - \frac{68000706738192229913544430210650205958978395980203253929201033243830931117492}{554969134813595233335922487728293913726901356879320194783918783565356953532967} a - \frac{23354102677004545794014747683491123162473856397024136003537013920871642652510}{184989711604531744445307495909431304575633785626440064927972927855118984510989}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times S_3$ (as 15T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 30
The 15 conjugacy class representatives for $S_3 \times C_5$
Character table for $S_3 \times C_5$

Intermediate fields

3.1.740652.1, 5.5.991199501189041.3

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ $15$ R $15$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ R ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ $15$ $15$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{5}$

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$2.15.10.1$x^{15} + 8 x^{6} + 32$$3$$5$$10$$S_3 \times C_5$$[\ ]_{3}^{10}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$11$11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$31$31.5.4.3$x^{5} - 1519$$5$$1$$4$$C_5$$[\ ]_{5}$
31.10.9.2$x^{10} - 1519$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$181$181.5.4.1$x^{5} - 181$$5$$1$$4$$C_5$$[\ ]_{5}$
181.10.9.8$x^{10} + 5792$$10$$1$$9$$C_{10}$$[\ ]_{10}$