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Properties

Label 15.1.71544884537...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{18}\cdot 3^{13}\cdot 5^{17}\cdot 11^{12}\cdot 59^{5}$
Root discriminant $977.92$
Ramified primes $2, 3, 5, 11, 59$
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![-50114084096, 21933571840, -5489655040, -438343680, -1020183680, 46028768, 27969920, -6949920, -1889940, -246940, 8327, 8365, 210, -130, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 130*x^13 + 210*x^12 + 8365*x^11 + 8327*x^10 - 246940*x^9 - 1889940*x^8 - 6949920*x^7 + 27969920*x^6 + 46028768*x^5 - 1020183680*x^4 - 438343680*x^3 - 5489655040*x^2 + 21933571840*x - 50114084096)
 
gp: K = bnfinit(x^15 - 5*x^14 - 130*x^13 + 210*x^12 + 8365*x^11 + 8327*x^10 - 246940*x^9 - 1889940*x^8 - 6949920*x^7 + 27969920*x^6 + 46028768*x^5 - 1020183680*x^4 - 438343680*x^3 - 5489655040*x^2 + 21933571840*x - 50114084096, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 130 x^{13} + 210 x^{12} + 8365 x^{11} + 8327 x^{10} - 246940 x^{9} - 1889940 x^{8} - 6949920 x^{7} + 27969920 x^{6} + 46028768 x^{5} - 1020183680 x^{4} - 438343680 x^{3} - 5489655040 x^{2} + 21933571840 x - 50114084096 \)

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:  \(-715448845372327674720280403400000000000000000=-\,2^{18}\cdot 3^{13}\cdot 5^{17}\cdot 11^{12}\cdot 59^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $977.92$
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:  $2, 3, 5, 11, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{5} - \frac{1}{2} a^{3} + \frac{2}{5}$, $\frac{1}{60} a^{6} - \frac{1}{20} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{2}{5} a + \frac{2}{15}$, $\frac{1}{60} a^{7} + \frac{1}{4} a^{3} + \frac{2}{5} a^{2} + \frac{1}{3} a$, $\frac{1}{120} a^{8} - \frac{1}{20} a^{5} + \frac{1}{8} a^{4} + \frac{9}{20} a^{3} - \frac{1}{3} a^{2} - \frac{1}{5}$, $\frac{1}{600} a^{9} + \frac{1}{600} a^{8} - \frac{1}{150} a^{7} - \frac{1}{150} a^{6} - \frac{3}{200} a^{5} + \frac{13}{200} a^{4} + \frac{41}{150} a^{3} + \frac{11}{150} a^{2} - \frac{7}{75} a - \frac{22}{75}$, $\frac{1}{3600} a^{10} - \frac{1}{1200} a^{9} - \frac{1}{450} a^{8} - \frac{1}{200} a^{7} - \frac{23}{3600} a^{6} - \frac{3}{80} a^{5} - \frac{221}{1800} a^{4} + \frac{49}{300} a^{3} + \frac{181}{450} a^{2} + \frac{12}{25} a + \frac{29}{225}$, $\frac{1}{7200} a^{11} - \frac{1}{7200} a^{10} - \frac{1}{3600} a^{9} - \frac{11}{3600} a^{8} + \frac{13}{7200} a^{7} + \frac{11}{7200} a^{6} + \frac{13}{360} a^{5} + \frac{43}{1800} a^{4} - \frac{163}{450} a^{3} + \frac{26}{225} a^{2} - \frac{187}{450} a + \frac{23}{225}$, $\frac{1}{72000} a^{12} + \frac{1}{24000} a^{11} + \frac{1}{12000} a^{10} - \frac{1}{2400} a^{9} + \frac{1}{1600} a^{8} - \frac{59}{24000} a^{7} - \frac{149}{18000} a^{6} + \frac{149}{6000} a^{5} - \frac{29}{150} a^{4} + \frac{137}{300} a^{3} + \frac{129}{500} a^{2} - \frac{317}{750} a + \frac{154}{1125}$, $\frac{1}{432000} a^{13} - \frac{1}{144000} a^{12} + \frac{7}{108000} a^{11} + \frac{7}{216000} a^{10} + \frac{53}{86400} a^{9} - \frac{607}{432000} a^{8} + \frac{1453}{216000} a^{7} - \frac{619}{108000} a^{6} + \frac{94}{3375} a^{5} - \frac{119}{2700} a^{4} - \frac{6559}{27000} a^{3} - \frac{527}{6750} a^{2} + \frac{41}{250} a - \frac{682}{3375}$, $\frac{1}{1701870273918881169128918371807986050352000} a^{14} - \frac{236876877963573791075785670113627699}{567290091306293723042972790602662016784000} a^{13} + \frac{72637843961200825764794751097556813}{850935136959440584564459185903993025176000} a^{12} - \frac{110747173949294023429552670225937031}{6807481095675524676515673487231944201408} a^{11} + \frac{43185002195224617729747574319210806913}{340374054783776233825783674361597210070400} a^{10} - \frac{656871960033959990923206716939112243877}{1701870273918881169128918371807986050352000} a^{9} - \frac{277211050264804316554527599459416294231}{106366892119930073070557398237999128147000} a^{8} + \frac{1319383369244635085307703577340317772887}{425467568479720292282229592951996512588000} a^{7} - \frac{45201750590074880944305364683546716713}{10636689211993007307055739823799912814700} a^{6} + \frac{1048169969912970446442549179282572957771}{42546756847972029228222959295199651258800} a^{5} - \frac{2375084718504424965227152258164690146408}{13295861514991259133819674779749891018375} a^{4} + \frac{6382566413733948527973550240301093165006}{13295861514991259133819674779749891018375} a^{3} + \frac{2155140521568624389434545215645840519953}{8863907676660839422546449853166594012250} a^{2} + \frac{353907366095438466333143913356930612242}{2659172302998251826763934955949978203675} a - \frac{224867534322995091537425753828531324144}{886390767666083942254644985316659401225}$

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:  \( 8791846785093799.0 \) (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

3.1.17700.1, 5.1.59296050000.9

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ R

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$$[\ ]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{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}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$