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Properties

Label 15.5.33713586078...0000.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{15}\cdot 5^{25}\cdot 11^{13}$
Root discriminant $233.63$
Ramified primes $2, 5, 11$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
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![7384846832, -2315335000, -1733605720, 762404060, -40668100, -70930618, 36375350, 6912345, -2425610, -442200, 25674, 6050, -330, -110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 110*x^13 - 330*x^12 + 6050*x^11 + 25674*x^10 - 442200*x^9 - 2425610*x^8 + 6912345*x^7 + 36375350*x^6 - 70930618*x^5 - 40668100*x^4 + 762404060*x^3 - 1733605720*x^2 - 2315335000*x + 7384846832)
 
gp: K = bnfinit(x^15 - 110*x^13 - 330*x^12 + 6050*x^11 + 25674*x^10 - 442200*x^9 - 2425610*x^8 + 6912345*x^7 + 36375350*x^6 - 70930618*x^5 - 40668100*x^4 + 762404060*x^3 - 1733605720*x^2 - 2315335000*x + 7384846832, 1)
 

Normalized defining polynomial

\( x^{15} - 110 x^{13} - 330 x^{12} + 6050 x^{11} + 25674 x^{10} - 442200 x^{9} - 2425610 x^{8} + 6912345 x^{7} + 36375350 x^{6} - 70930618 x^{5} - 40668100 x^{4} + 762404060 x^{3} - 1733605720 x^{2} - 2315335000 x + 7384846832 \)

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:  \(-337135860780576171875000000000000000=-\,2^{15}\cdot 5^{25}\cdot 11^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $233.63$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{16} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{3520} a^{10} + \frac{1}{64} a^{9} - \frac{1}{32} a^{8} - \frac{1}{16} a^{7} - \frac{3}{64} a^{6} - \frac{69}{320} a^{5} + \frac{1}{32} a^{4} + \frac{1}{4} a^{3} + \frac{5}{16} a^{2} - \frac{3}{16} a - \frac{7}{40}$, $\frac{1}{7040} a^{11} - \frac{1}{128} a^{9} + \frac{1}{64} a^{8} + \frac{9}{128} a^{7} + \frac{9}{160} a^{6} + \frac{1}{128} a^{5} - \frac{11}{64} a^{4} - \frac{3}{32} a^{3} - \frac{5}{16} a^{2} - \frac{69}{160} a - \frac{3}{16}$, $\frac{1}{7040} a^{12} - \frac{1}{7040} a^{10} - \frac{3}{128} a^{8} - \frac{1}{160} a^{7} + \frac{15}{128} a^{6} + \frac{11}{160} a^{5} + \frac{1}{8} a^{4} + \frac{7}{16} a^{3} + \frac{1}{160} a^{2} - \frac{1}{2} a - \frac{9}{40}$, $\frac{1}{535040} a^{13} + \frac{17}{267520} a^{12} - \frac{3}{535040} a^{11} - \frac{37}{267520} a^{10} + \frac{127}{9728} a^{9} - \frac{1347}{24320} a^{8} - \frac{2951}{48640} a^{7} + \frac{2001}{24320} a^{6} + \frac{4343}{24320} a^{5} + \frac{353}{2432} a^{4} + \frac{371}{12160} a^{3} - \frac{273}{6080} a^{2} + \frac{1011}{6080} a + \frac{229}{3040}$, $\frac{1}{154485165875657947796413667219559037897523855360} a^{14} + \frac{1675892671493494930689186984048260595823}{3592678276178091809218922493478117160407531520} a^{13} + \frac{7020315331514586351596467142270239284043243}{154485165875657947796413667219559037897523855360} a^{12} - \frac{9013251200039932622482110991857090267886291}{154485165875657947796413667219559037897523855360} a^{11} - \frac{8927100271426085953278565357041248018776157}{154485165875657947796413667219559037897523855360} a^{10} - \frac{155375405436574440911854273442489335982529109}{14044105988696177072401242474505367081593077760} a^{9} + \frac{652263402218777272977352138737089593488617519}{14044105988696177072401242474505367081593077760} a^{8} - \frac{80309569235434939328607951277592241374337793}{739163473089272477494802235500282477978583040} a^{7} + \frac{20663641519167939826153047549356203497223131}{3511026497174044268100310618626341770398269440} a^{6} - \frac{82420416809230694286510616047408747866374401}{7022052994348088536200621237252683540796538880} a^{5} - \frac{27800519362659069768148571167535923046988407}{1755513248587022134050155309313170885199134720} a^{4} - \frac{911679060518210862221250954344200920853427161}{3511026497174044268100310618626341770398269440} a^{3} - \frac{158493925322916314071305782716253585304772859}{438878312146755533512538827328292721299783680} a^{2} - \frac{347357213130482929356134548836593579952806013}{1755513248587022134050155309313170885199134720} a - \frac{1077525738951396181886146714863920684374571}{20412944751011885279652968712943847502315520}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  \( 36521311280903.49 \) (assuming GRH)
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

3.1.440.1, 5.5.5719140625.4

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 $15$ R $15$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ $15$ $15$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
$5$5.5.8.4$x^{5} - 5 x^{4} + 55$$5$$1$$8$$C_5$$[2]$
5.10.17.27$x^{10} - 10 x^{8} + 10$$10$$1$$17$$C_{10}$$[2]_{2}$
$11$11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.2$x^{10} - 72171$$10$$1$$9$$C_{10}$$[\ ]_{10}$