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Properties

Label 15.1.74813147538...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 5^{17}\cdot 7^{13}\cdot 11^{13}\cdot 31^{5}$
Root discriminant $1333.31$
Ramified primes $2, 5, 7, 11, 31$
Class number $150$ (GRH)
Class group $[5, 30]$ (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![-28950040844263, 0, 0, 0, 0, 18380656217, 0, 0, 0, 0, -85701, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 85701*x^10 + 18380656217*x^5 - 28950040844263)
 
gp: K = bnfinit(x^15 - 85701*x^10 + 18380656217*x^5 - 28950040844263, 1)
 

Normalized defining polynomial

\( x^{15} - 85701 x^{10} + 18380656217 x^{5} - 28950040844263 \)

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:  \(-74813147538844994892441022008177343750000000000=-\,2^{10}\cdot 5^{17}\cdot 7^{13}\cdot 11^{13}\cdot 31^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1333.31$
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:  $2, 5, 7, 11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{7} a^{4}$, $\frac{1}{245} a^{5} + \frac{2}{5}$, $\frac{1}{245} a^{6} + \frac{2}{5} a$, $\frac{1}{1715} a^{7} - \frac{13}{35} a^{2}$, $\frac{1}{12005} a^{8} - \frac{13}{245} a^{3}$, $\frac{1}{420175} a^{9} + \frac{1}{60025} a^{8} + \frac{1}{8575} a^{7} + \frac{1}{1225} a^{6} + \frac{2}{1225} a^{5} - \frac{83}{8575} a^{4} - \frac{83}{1225} a^{3} - \frac{83}{175} a^{2} - \frac{8}{25} a + \frac{9}{25}$, $\frac{1}{38856523475} a^{10} - \frac{13781}{72090025} a^{5} + \frac{1401}{30025}$, $\frac{1}{271995664325} a^{11} + \frac{574709}{504630175} a^{6} - \frac{4947}{30025} a$, $\frac{1}{104718330765125} a^{12} - \frac{2}{1359978321625} a^{11} + \frac{1}{194282617375} a^{10} + \frac{8813569}{194282617375} a^{7} + \frac{2970012}{2523150875} a^{6} - \frac{13781}{360450125} a^{5} + \frac{4937168}{11559625} a^{2} + \frac{3889}{150125} a - \frac{28624}{150125}$, $\frac{1}{733028315355875} a^{13} + \frac{2}{1359978321625} a^{11} + \frac{2}{194282617375} a^{10} + \frac{8813569}{1359978321625} a^{8} - \frac{2970012}{2523150875} a^{6} - \frac{27562}{360450125} a^{5} + \frac{4937168}{80917375} a^{3} - \frac{3889}{150125} a - \frac{57248}{150125}$, $\frac{1}{56443180282402375} a^{14} + \frac{1}{1359978321625} a^{11} - \frac{2}{194282617375} a^{10} + \frac{99441029}{104718330765125} a^{9} - \frac{1}{60025} a^{8} - \frac{1}{8575} a^{7} - \frac{3544721}{2523150875} a^{6} - \frac{560928}{360450125} a^{5} + \frac{306676408}{6230637875} a^{4} + \frac{83}{1225} a^{3} + \frac{83}{175} a^{2} - \frac{28967}{150125} a + \frac{3203}{150125}$

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

Class group and class number

$C_{5}\times C_{30}$, which has order $150$ (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:  \( 4911429110107489.0 \) (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

3.1.238700.1, 5.1.109853253125.1

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 R R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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}$
$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$$[5/4]_{12}^{2}$
$7$7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
7.10.9.1$x^{10} - 7$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.5$x^{10} - 8019$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$