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Properties

Label 15.1.34898680216...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 3^{12}\cdot 5^{17}\cdot 29^{13}$
Root discriminant $799.57$
Ramified primes $2, 3, 5, 29$
Class number $50$ (GRH)
Class group $[5, 10]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8065689292, 2563933460, -2416077960, 592975880, -189916220, -2572288, 7674660, -3835960, -273620, 7380, -2819, 965, 10, -30, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 30*x^13 + 10*x^12 + 965*x^11 - 2819*x^10 + 7380*x^9 - 273620*x^8 - 3835960*x^7 + 7674660*x^6 - 2572288*x^5 - 189916220*x^4 + 592975880*x^3 - 2416077960*x^2 + 2563933460*x - 8065689292)
 
gp: K = bnfinit(x^15 - 5*x^14 - 30*x^13 + 10*x^12 + 965*x^11 - 2819*x^10 + 7380*x^9 - 273620*x^8 - 3835960*x^7 + 7674660*x^6 - 2572288*x^5 - 189916220*x^4 + 592975880*x^3 - 2416077960*x^2 + 2563933460*x - 8065689292, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 30 x^{13} + 10 x^{12} + 965 x^{11} - 2819 x^{10} + 7380 x^{9} - 273620 x^{8} - 3835960 x^{7} + 7674660 x^{6} - 2572288 x^{5} - 189916220 x^{4} + 592975880 x^{3} - 2416077960 x^{2} + 2563933460 x - 8065689292 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-34898680216597968037915833600000000000000000=-\,2^{23}\cdot 3^{12}\cdot 5^{17}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $799.57$
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:  $2, 3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{2}{5}$, $\frac{1}{15} a^{6} + \frac{1}{15} a^{5} + \frac{1}{3} a^{3} + \frac{7}{15} a + \frac{7}{15}$, $\frac{1}{15} a^{7} - \frac{1}{15} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{7}{15} a^{2} - \frac{7}{15}$, $\frac{1}{15} a^{8} - \frac{1}{3} a^{4} - \frac{1}{5} a^{3} - \frac{1}{3}$, $\frac{1}{75} a^{9} - \frac{2}{75} a^{8} - \frac{1}{75} a^{7} + \frac{2}{75} a^{6} + \frac{1}{75} a^{5} + \frac{17}{75} a^{4} - \frac{3}{25} a^{3} + \frac{8}{75} a^{2} + \frac{3}{25} a - \frac{8}{75}$, $\frac{1}{8700} a^{10} - \frac{31}{8700} a^{9} - \frac{7}{2175} a^{8} - \frac{39}{1450} a^{7} + \frac{173}{8700} a^{6} - \frac{229}{2900} a^{5} - \frac{988}{2175} a^{4} - \frac{844}{2175} a^{3} + \frac{418}{2175} a^{2} - \frac{389}{1450} a - \frac{503}{1450}$, $\frac{1}{8700} a^{11} + \frac{11}{1740} a^{9} - \frac{1}{30} a^{8} - \frac{1}{1740} a^{7} - \frac{49}{2175} a^{6} + \frac{49}{580} a^{5} - \frac{187}{435} a^{4} + \frac{181}{435} a^{3} + \frac{53}{290} a^{2} - \frac{133}{725} a - \frac{91}{290}$, $\frac{1}{130500} a^{12} - \frac{1}{130500} a^{11} + \frac{1}{32625} a^{10} - \frac{31}{6525} a^{9} - \frac{307}{26100} a^{8} + \frac{2579}{130500} a^{7} + \frac{869}{32625} a^{6} + \frac{433}{65250} a^{5} - \frac{2152}{6525} a^{4} - \frac{1847}{13050} a^{3} + \frac{18527}{65250} a^{2} - \frac{13561}{32625} a - \frac{3671}{32625}$, $\frac{1}{130500} a^{13} + \frac{1}{43500} a^{11} - \frac{1}{130500} a^{10} - \frac{17}{6525} a^{9} + \frac{247}{10875} a^{8} - \frac{427}{26100} a^{7} - \frac{4103}{130500} a^{6} - \frac{11}{14500} a^{5} + \frac{3797}{13050} a^{4} + \frac{12806}{32625} a^{3} + \frac{1861}{4350} a^{2} - \frac{10603}{21750} a - \frac{9697}{65250}$, $\frac{1}{5868157741347275803939371843265162209000} a^{14} - \frac{7939280318373886252165350282919687}{5868157741347275803939371843265162209000} a^{13} - \frac{491994756619013757079349350606259}{195605258044909193464645728108838740300} a^{12} + \frac{166205256062791872220290925052657713}{2934078870673637901969685921632581104500} a^{11} - \frac{1956222771797070666113285863381979}{391210516089818386929291456217677480600} a^{10} - \frac{10318851833032590600211857507698851091}{5868157741347275803939371843265162209000} a^{9} + \frac{93298329052676613372256989173504550821}{2934078870673637901969685921632581104500} a^{8} - \frac{417910855087528193012637499151780481}{32600876340818198910774288018139790050} a^{7} - \frac{49072250470598792445598475951858165333}{2934078870673637901969685921632581104500} a^{6} + \frac{1463095105660642654621277679071282929}{20235026694300951048066799459535042100} a^{5} - \frac{17361738887522430381847055228770521227}{326008763408181989107742880181397900500} a^{4} - \frac{81795607400064133487435688986945751917}{1467039435336818950984842960816290552250} a^{3} - \frac{1824348183957291235352927691621799103}{16300438170409099455387144009069895025} a^{2} + \frac{174952650730307994418673654371001575891}{1467039435336818950984842960816290552250} a - \frac{21880859999781768756746006490744765586}{146703943533681895098484296081629055225}$

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

Class group and class number

$C_{5}\times C_{10}$, which has order $50$ (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:  $7$
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:  \( 567390527898541.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.5800.1, 5.1.2864488050000.5

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

Sibling fields

Degree 30 siblings: 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.19.49$x^{10} - 6$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.15.17.3$x^{15} - 5 x^{13} + 10 x^{12} + 10 x^{11} + 10 x^{9} - 5 x^{8} - 10 x^{6} + 10 x^{5} + 5 x^{3} + 10$$15$$1$$17$$F_5 \times S_3$$[5/4]_{12}^{2}$
$29$29.5.4.1$x^{5} - 29$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
29.10.9.2$x^{10} + 58$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$