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Properties

Label 15.1.49623057464...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 5^{9}\cdot 13^{13}$
Root discriminant $70.21$
Ramified primes $2, 5, 13$
Class number $1$ (GRH)
Class group Trivial (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![-24862, 3820, -37650, 12520, -13630, 1367, -560, -4795, -450, 30, -38, 10, -10, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^13 - 10*x^12 + 10*x^11 - 38*x^10 + 30*x^9 - 450*x^8 - 4795*x^7 - 560*x^6 + 1367*x^5 - 13630*x^4 + 12520*x^3 - 37650*x^2 + 3820*x - 24862)
 
gp: K = bnfinit(x^15 - 5*x^13 - 10*x^12 + 10*x^11 - 38*x^10 + 30*x^9 - 450*x^8 - 4795*x^7 - 560*x^6 + 1367*x^5 - 13630*x^4 + 12520*x^3 - 37650*x^2 + 3820*x - 24862, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{13} - 10 x^{12} + 10 x^{11} - 38 x^{10} + 30 x^{9} - 450 x^{8} - 4795 x^{7} - 560 x^{6} + 1367 x^{5} - 13630 x^{4} + 12520 x^{3} - 37650 x^{2} + 3820 x - 24862 \)

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:  \(-4962305746407473152000000000=-\,2^{23}\cdot 5^{9}\cdot 13^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.21$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{26} a^{10} + \frac{11}{26} a^{9} + \frac{2}{13} a^{7} + \frac{7}{26} a^{6} - \frac{1}{26} a^{5} + \frac{2}{13} a^{4} - \frac{2}{13} a^{3} - \frac{1}{13} a + \frac{4}{13}$, $\frac{1}{26} a^{11} + \frac{9}{26} a^{9} + \frac{2}{13} a^{8} - \frac{11}{26} a^{7} - \frac{11}{26} a^{5} + \frac{2}{13} a^{4} - \frac{4}{13} a^{3} - \frac{1}{13} a^{2} + \frac{2}{13} a - \frac{5}{13}$, $\frac{1}{4030} a^{12} - \frac{1}{2015} a^{11} + \frac{11}{2015} a^{10} + \frac{1637}{4030} a^{9} + \frac{1983}{4030} a^{8} - \frac{262}{2015} a^{7} - \frac{142}{2015} a^{6} - \frac{1}{10} a^{5} + \frac{538}{2015} a^{4} - \frac{227}{2015} a^{3} - \frac{919}{2015} a^{2} + \frac{43}{2015} a + \frac{28}{65}$, $\frac{1}{4030} a^{13} + \frac{9}{2015} a^{11} - \frac{12}{2015} a^{10} - \frac{704}{2015} a^{9} - \frac{294}{2015} a^{8} - \frac{46}{2015} a^{7} - \frac{408}{2015} a^{6} + \frac{395}{806} a^{5} - \frac{42}{155} a^{4} + \frac{22}{2015} a^{3} + \frac{44}{403} a^{2} + \frac{644}{2015} a + \frac{31}{65}$, $\frac{1}{37069325189501114284580} a^{14} - \frac{1224076015350913689}{37069325189501114284580} a^{13} + \frac{384655082837912213}{18534662594750557142290} a^{12} + \frac{323157236927455684249}{18534662594750557142290} a^{11} + \frac{68696078829378789626}{9267331297375278571145} a^{10} + \frac{1035923308686609121989}{3706932518950111428458} a^{9} - \frac{3704149316201931263643}{18534662594750557142290} a^{8} - \frac{278833608242392124165}{3706932518950111428458} a^{7} - \frac{1107836254625249996753}{37069325189501114284580} a^{6} + \frac{2064766165551803202089}{37069325189501114284580} a^{5} + \frac{875360065223750661189}{3706932518950111428458} a^{4} - \frac{2525054185709318169302}{9267331297375278571145} a^{3} + \frac{2084019527881995964756}{9267331297375278571145} a^{2} + \frac{616176631796131621923}{1425743276519273626330} a - \frac{29142451077064474083}{119578468353229400918}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 228104823.0554214 \) (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

3.1.104.1, 5.1.57122000.1

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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.5.4.1$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}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$13$13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
13.10.9.2$x^{10} + 26$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$