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Properties

Label 15.1.21938743571...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{15}\cdot 5^{17}\cdot 11^{13}\cdot 191^{5}$
Root discriminant $570.27$
Ramified primes $2, 5, 11, 191$
Class number $25$ (GRH)
Class group $[5, 5]$ (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![1953811463264, -1460294354910, 402491806260, -11485832055, -19651259480, 4295896648, -8391805, -119368045, 14288760, 818485, -292788, 11465, 2285, -205, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 205*x^13 + 2285*x^12 + 11465*x^11 - 292788*x^10 + 818485*x^9 + 14288760*x^8 - 119368045*x^7 - 8391805*x^6 + 4295896648*x^5 - 19651259480*x^4 - 11485832055*x^3 + 402491806260*x^2 - 1460294354910*x + 1953811463264)
 
gp: K = bnfinit(x^15 - 5*x^14 - 205*x^13 + 2285*x^12 + 11465*x^11 - 292788*x^10 + 818485*x^9 + 14288760*x^8 - 119368045*x^7 - 8391805*x^6 + 4295896648*x^5 - 19651259480*x^4 - 11485832055*x^3 + 402491806260*x^2 - 1460294354910*x + 1953811463264, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 205 x^{13} + 2285 x^{12} + 11465 x^{11} - 292788 x^{10} + 818485 x^{9} + 14288760 x^{8} - 119368045 x^{7} - 8391805 x^{6} + 4295896648 x^{5} - 19651259480 x^{4} - 11485832055 x^{3} + 402491806260 x^{2} - 1460294354910 x + 1953811463264 \)

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:  \(-219387435712728438617734525000000000000000=-\,2^{15}\cdot 5^{17}\cdot 11^{13}\cdot 191^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $570.27$
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:  $2, 5, 11, 191$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{9} + \frac{1}{25} a^{8} + \frac{1}{25} a^{7} + \frac{1}{25} a^{6} + \frac{1}{25} a^{5} + \frac{9}{25} a^{4} + \frac{9}{25} a^{3} + \frac{9}{25} a^{2} + \frac{9}{25} a + \frac{9}{25}$, $\frac{1}{550} a^{10} + \frac{1}{55} a^{9} - \frac{7}{110} a^{8} + \frac{3}{55} a^{7} - \frac{2}{55} a^{6} - \frac{1}{25} a^{5} + \frac{19}{55} a^{4} + \frac{26}{55} a^{3} + \frac{19}{110} a^{2} - \frac{8}{55} a - \frac{127}{275}$, $\frac{1}{550} a^{11} - \frac{3}{550} a^{9} - \frac{19}{275} a^{8} + \frac{16}{275} a^{7} - \frac{2}{55} a^{6} - \frac{4}{275} a^{5} + \frac{49}{275} a^{4} - \frac{217}{550} a^{3} - \frac{31}{275} a^{2} - \frac{68}{275} a - \frac{61}{275}$, $\frac{1}{2750} a^{12} - \frac{1}{1375} a^{11} + \frac{1}{2750} a^{10} + \frac{3}{275} a^{9} - \frac{12}{275} a^{8} - \frac{26}{1375} a^{7} + \frac{97}{1375} a^{6} + \frac{24}{1375} a^{5} - \frac{221}{550} a^{4} + \frac{117}{275} a^{3} - \frac{212}{1375} a^{2} + \frac{454}{1375} a - \frac{12}{1375}$, $\frac{1}{2750} a^{13} + \frac{1}{1375} a^{11} + \frac{1}{1375} a^{10} - \frac{9}{550} a^{9} - \frac{101}{1375} a^{8} + \frac{23}{275} a^{7} + \frac{83}{1375} a^{6} - \frac{59}{2750} a^{5} - \frac{53}{275} a^{4} - \frac{149}{2750} a^{3} - \frac{68}{275} a^{2} - \frac{334}{1375} a - \frac{534}{1375}$, $\frac{1}{12180672305688935383701745050904611954581748730150250} a^{14} + \frac{5311324913707693697155740709050976981276646627}{2436134461137787076740349010180922390916349746030050} a^{13} + \frac{86258504995572984097964692069315788671551957706}{1218067230568893538370174505090461195458174873015025} a^{12} + \frac{9991200918951068527026780414012004283126889097711}{12180672305688935383701745050904611954581748730150250} a^{11} + \frac{8695142117110920580759034316256474402668944028663}{12180672305688935383701745050904611954581748730150250} a^{10} + \frac{120657249857734499387949254463515277422338214716609}{6090336152844467691850872525452305977290874365075125} a^{9} + \frac{27997666164938580381144195079929530954485470086003}{1218067230568893538370174505090461195458174873015025} a^{8} - \frac{94494398269609666033619459842860178147264922760302}{1218067230568893538370174505090461195458174873015025} a^{7} - \frac{1050117169817774215234119787952182022029608249276527}{12180672305688935383701745050904611954581748730150250} a^{6} - \frac{33330371668142154956853976078327607697616632149281}{1107333845971721398518340459173146541325613520922750} a^{5} - \frac{4397546740696003805272953266455516595337044324844119}{12180672305688935383701745050904611954581748730150250} a^{4} + \frac{402240717620987323519691259528836796910191729593546}{1218067230568893538370174505090461195458174873015025} a^{3} + \frac{309673264525681046818442426368949137514715338679671}{1218067230568893538370174505090461195458174873015025} a^{2} - \frac{2234823280806746250351870500649093310491909510994292}{6090336152844467691850872525452305977290874365075125} a - \frac{2333682772596455315714807195004479253660554538024886}{6090336152844467691850872525452305977290874365075125}$

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:  $7$
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:  \( 80177251366909.72 \) (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

3.1.420200.1, 5.1.45753125.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{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$$[5/4]_{12}^{2}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$191$$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
191.2.1.1$x^{2} - 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.1$x^{2} - 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.1$x^{2} - 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.1$x^{2} - 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.1$x^{2} - 191$$2$$1$$1$$C_2$$[\ ]_{2}$