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Properties

Label 15.7.63070221167...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 251^{4}\cdot 2477^{2}\cdot 473117^{2}$
Root discriminant $1130.63$
Ramified primes $2, 5, 7, 251, 2477, 473117$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T92

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1028096000, -3598336000, 2403174400, 2618432000, -2313216000, -260002560, 521764864, -80152416, -12871296, 1327176, 371872, 15004, -2736, -360, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 360*x^13 - 2736*x^12 + 15004*x^11 + 371872*x^10 + 1327176*x^9 - 12871296*x^8 - 80152416*x^7 + 521764864*x^6 - 260002560*x^5 - 2313216000*x^4 + 2618432000*x^3 + 2403174400*x^2 - 3598336000*x + 1028096000)
 
gp: K = bnfinit(x^15 - 360*x^13 - 2736*x^12 + 15004*x^11 + 371872*x^10 + 1327176*x^9 - 12871296*x^8 - 80152416*x^7 + 521764864*x^6 - 260002560*x^5 - 2313216000*x^4 + 2618432000*x^3 + 2403174400*x^2 - 3598336000*x + 1028096000, 1)
 

Normalized defining polynomial

\( x^{15} - 360 x^{13} - 2736 x^{12} + 15004 x^{11} + 371872 x^{10} + 1327176 x^{9} - 12871296 x^{8} - 80152416 x^{7} + 521764864 x^{6} - 260002560 x^{5} - 2313216000 x^{4} + 2618432000 x^{3} + 2403174400 x^{2} - 3598336000 x + 1028096000 \)

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:  \(6307022116728638404825373947273030733824000000=2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 251^{4}\cdot 2477^{2}\cdot 473117^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1130.63$
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:  $2, 5, 7, 251, 2477, 473117$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{16} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{8} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{9} - \frac{1}{16} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{128} a^{10} - \frac{1}{32} a^{6} + \frac{1}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{1280} a^{11} - \frac{1}{80} a^{8} - \frac{9}{320} a^{7} + \frac{1}{40} a^{6} - \frac{3}{160} a^{5} + \frac{1}{20} a^{4} + \frac{7}{40} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{2560} a^{12} - \frac{1}{160} a^{9} - \frac{9}{640} a^{8} + \frac{1}{80} a^{7} - \frac{3}{320} a^{6} - \frac{1}{10} a^{5} + \frac{7}{80} a^{4} - \frac{1}{10} a^{3}$, $\frac{1}{51200} a^{13} + \frac{1}{800} a^{10} - \frac{49}{12800} a^{9} - \frac{9}{1600} a^{8} - \frac{123}{6400} a^{7} - \frac{9}{800} a^{6} + \frac{67}{1600} a^{5} - \frac{7}{400} a^{4} - \frac{9}{40} a^{3} + \frac{1}{5} a^{2} + \frac{1}{4} a$, $\frac{1}{271897504549611157775143027164051167679260876800} a^{14} + \frac{530170842986547307859685105250055197601653}{67974376137402789443785756791012791919815219200} a^{13} + \frac{532356817264359571657041288402039519685633}{6797437613740278944378575679101279191981521920} a^{12} + \frac{4559714001984181518715553714925829145335079}{16993594034350697360946439197753197979953804800} a^{11} + \frac{111267658427118235534380765910001434690738343}{67974376137402789443785756791012791919815219200} a^{10} + \frac{8878064105908523360302535591769030505344153}{3398718806870139472189287839550639595990760960} a^{9} + \frac{100754011619101299379247846792490476192330141}{6797437613740278944378575679101279191981521920} a^{8} - \frac{185133774080013782818933957184830514648897967}{8496797017175348680473219598876598989976902400} a^{7} + \frac{287393863872157137638308314127137839123080561}{8496797017175348680473219598876598989976902400} a^{6} - \frac{239742194941633144675566510017207301735196171}{2124199254293837170118304899719149747494225600} a^{5} + \frac{2274848210616672234306369381110823901377851}{24699991328998106629282615113013369156909600} a^{4} + \frac{1394044800442001597719386013530295572955509}{53104981357345929252957622492978743687355640} a^{3} - \frac{26011220436987506902217873425537172316973707}{106209962714691858505915244985957487374711280} a^{2} - \frac{749380617661966158303192432212376295367311}{2655249067867296462647881124648937184367782} a - \frac{637587889690411770177933373535873937048738}{1327624533933648231323940562324468592183891}$

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:  \( 153449554548000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T92:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 648000
The 55 conjugacy class representatives for [A(5)^3]3=A(5)wr3 are not computed
Character table for [A(5)^3]3=A(5)wr3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 sibling: data not computed
Degree 36 sibling: 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 R $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $15$ $15$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[2, 2, 3]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
251Data not computed
2477Data not computed
473117Data not computed