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Properties

Label 15.1.14430641089...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 3^{13}\cdot 5^{17}\cdot 41^{5}$
Root discriminant $87.89$
Ramified primes $2, 3, 5, 41$
Class number $3$ (GRH)
Class group $[3]$ (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![-84035, -120050, -162925, 145530, 89950, 40325, -38665, -11100, -2970, 4235, 650, 250, -120, 5, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 5*x^13 - 120*x^12 + 250*x^11 + 650*x^10 + 4235*x^9 - 2970*x^8 - 11100*x^7 - 38665*x^6 + 40325*x^5 + 89950*x^4 + 145530*x^3 - 162925*x^2 - 120050*x - 84035)
 
gp: K = bnfinit(x^15 - 5*x^14 + 5*x^13 - 120*x^12 + 250*x^11 + 650*x^10 + 4235*x^9 - 2970*x^8 - 11100*x^7 - 38665*x^6 + 40325*x^5 + 89950*x^4 + 145530*x^3 - 162925*x^2 - 120050*x - 84035, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 5 x^{13} - 120 x^{12} + 250 x^{11} + 650 x^{10} + 4235 x^{9} - 2970 x^{8} - 11100 x^{7} - 38665 x^{6} + 40325 x^{5} + 89950 x^{4} + 145530 x^{3} - 162925 x^{2} - 120050 x - 84035 \)

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:  \(-144306410896033593750000000000=-\,2^{10}\cdot 3^{13}\cdot 5^{17}\cdot 41^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.89$
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:  $2, 3, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{8} - \frac{1}{3} a^{6} - \frac{1}{6} a^{4} + \frac{1}{3} a^{2} - \frac{1}{6}$, $\frac{1}{42} a^{11} + \frac{1}{21} a^{10} + \frac{19}{42} a^{9} + \frac{10}{21} a^{8} - \frac{1}{21} a^{7} + \frac{10}{21} a^{6} - \frac{1}{6} a^{5} - \frac{1}{21} a^{4} + \frac{1}{21} a^{3} + \frac{5}{21} a^{2} + \frac{5}{42} a - \frac{1}{3}$, $\frac{1}{294} a^{12} + \frac{1}{147} a^{11} + \frac{19}{294} a^{10} + \frac{31}{147} a^{9} - \frac{1}{147} a^{8} + \frac{73}{147} a^{7} - \frac{19}{42} a^{6} - \frac{64}{147} a^{5} - \frac{20}{147} a^{4} + \frac{5}{147} a^{3} - \frac{79}{294} a^{2} + \frac{5}{21} a$, $\frac{1}{2058} a^{13} + \frac{1}{1029} a^{12} + \frac{19}{2058} a^{11} + \frac{13}{2058} a^{10} - \frac{1}{1029} a^{9} + \frac{979}{2058} a^{8} + \frac{65}{294} a^{7} + \frac{93}{343} a^{6} - \frac{167}{1029} a^{5} - \frac{529}{2058} a^{4} - \frac{79}{2058} a^{3} + \frac{40}{147} a^{2} + \frac{2}{7} a + \frac{1}{6}$, $\frac{1}{309354006181947033741530099598} a^{14} - \frac{38238760662507593133849055}{309354006181947033741530099598} a^{13} + \frac{6240426577808467296013567}{3772609831487158948067440239} a^{12} - \frac{854425118334179029358273956}{154677003090973516870765049799} a^{11} + \frac{4350329651681875511420963485}{154677003090973516870765049799} a^{10} - \frac{108480728767453363888061373929}{309354006181947033741530099598} a^{9} - \frac{634110446940895014059780641}{2455190525253547886837540473} a^{8} - \frac{3725557994396623560304869643}{34372667353549670415725566622} a^{7} - \frac{16185471918528973267109369695}{103118002060649011247176699866} a^{6} + \frac{12082297133929760953942889459}{309354006181947033741530099598} a^{5} - \frac{652834403599915767841541720}{3772609831487158948067440239} a^{4} - \frac{9990134069578814805821986027}{44193429454563861963075728514} a^{3} + \frac{2125484962616926709148968879}{6313347064937694566153675502} a^{2} - \frac{142798297283097993218886065}{901906723562527795164810786} a - \frac{12918473745590761264311601}{128843817651789685023544398}$

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:  $7$
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:  \( 545829701.3427438 \) (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

3.1.12300.1, 5.1.253125.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{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$$[5/4]_{12}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$