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Properties

Label 15.1.29997949942...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{12}\cdot 5^{10}\cdot 13^{13}\cdot 19^{5}$
Root discriminant $125.45$
Ramified primes $2, 5, 13, 19$
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![364550, -154970, 382390, -328340, 143020, -32349, 84935, -49345, 6270, 3015, -1273, -95, 140, -15, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 15*x^13 + 140*x^12 - 95*x^11 - 1273*x^10 + 3015*x^9 + 6270*x^8 - 49345*x^7 + 84935*x^6 - 32349*x^5 + 143020*x^4 - 328340*x^3 + 382390*x^2 - 154970*x + 364550)
 
gp: K = bnfinit(x^15 - 5*x^14 - 15*x^13 + 140*x^12 - 95*x^11 - 1273*x^10 + 3015*x^9 + 6270*x^8 - 49345*x^7 + 84935*x^6 - 32349*x^5 + 143020*x^4 - 328340*x^3 + 382390*x^2 - 154970*x + 364550, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 15 x^{13} + 140 x^{12} - 95 x^{11} - 1273 x^{10} + 3015 x^{9} + 6270 x^{8} - 49345 x^{7} + 84935 x^{6} - 32349 x^{5} + 143020 x^{4} - 328340 x^{3} + 382390 x^{2} - 154970 x + 364550 \)

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:  \(-29997949942318842441880000000000=-\,2^{12}\cdot 5^{10}\cdot 13^{13}\cdot 19^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $125.45$
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:  $2, 5, 13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{4}$, $\frac{1}{65} a^{10} + \frac{3}{65} a^{9} - \frac{1}{65} a^{8} + \frac{1}{65} a^{7} - \frac{21}{65} a^{6} + \frac{12}{65} a^{5} - \frac{28}{65} a^{4} - \frac{6}{13} a^{3} - \frac{5}{13} a^{2} - \frac{6}{13} a + \frac{6}{13}$, $\frac{1}{65} a^{11} + \frac{3}{65} a^{9} + \frac{4}{65} a^{8} - \frac{24}{65} a^{7} + \frac{2}{13} a^{6} + \frac{1}{65} a^{5} + \frac{2}{65} a^{4} - \frac{4}{13} a^{2} - \frac{2}{13} a - \frac{5}{13}$, $\frac{1}{325} a^{12} + \frac{2}{325} a^{11} + \frac{1}{325} a^{10} + \frac{6}{65} a^{9} + \frac{1}{13} a^{8} - \frac{79}{325} a^{7} + \frac{37}{325} a^{6} + \frac{71}{325} a^{5} + \frac{12}{65} a^{4} - \frac{31}{65} a^{3} + \frac{16}{65} a + \frac{6}{13}$, $\frac{1}{325} a^{13} + \frac{2}{325} a^{11} - \frac{2}{325} a^{10} + \frac{4}{65} a^{9} - \frac{14}{325} a^{8} - \frac{4}{65} a^{7} + \frac{92}{325} a^{6} + \frac{148}{325} a^{5} + \frac{24}{65} a^{4} - \frac{18}{65} a^{3} + \frac{16}{65} a^{2} - \frac{27}{65} a - \frac{1}{13}$, $\frac{1}{2699138904882592696818967020725} a^{14} + \frac{675805552653382912619513074}{539827780976518539363793404145} a^{13} + \frac{3252303361775935662159141506}{2699138904882592696818967020725} a^{12} + \frac{20746920678914104814561408551}{2699138904882592696818967020725} a^{11} - \frac{14622604979386432304184077426}{2699138904882592696818967020725} a^{10} - \frac{213331261518171519586377978609}{2699138904882592696818967020725} a^{9} - \frac{20573006671483993323515455191}{539827780976518539363793404145} a^{8} + \frac{852424632097731667923730067731}{2699138904882592696818967020725} a^{7} - \frac{69472095281476217582923929753}{207626069606353284370689770825} a^{6} - \frac{55145725597784973295795984691}{2699138904882592696818967020725} a^{5} + \frac{91113392142041374743828140356}{539827780976518539363793404145} a^{4} - \frac{155261961474571632746789336768}{539827780976518539363793404145} a^{3} - \frac{19416786289033583920858121467}{539827780976518539363793404145} a^{2} - \frac{141150415715913554160896004691}{539827780976518539363793404145} a - \frac{44997829385140682521081002955}{107965556195303707872758680829}$

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:  \( 17265952420.66845 \) (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

3.1.1235.1, 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 ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
13.10.9.1$x^{10} - 13$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$