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Properties

Label 15.3.15330693091...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{12}\cdot 3^{13}\cdot 5^{17}\cdot 79^{5}$
Root discriminant $119.96$
Ramified primes $2, 3, 5, 79$
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![54880, -215600, 14000, 133680, -439550, 418175, -261205, 177525, -44010, 34115, -4825, 2665, -270, 95, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 95*x^13 - 270*x^12 + 2665*x^11 - 4825*x^10 + 34115*x^9 - 44010*x^8 + 177525*x^7 - 261205*x^6 + 418175*x^5 - 439550*x^4 + 133680*x^3 + 14000*x^2 - 215600*x + 54880)
 
gp: K = bnfinit(x^15 - 5*x^14 + 95*x^13 - 270*x^12 + 2665*x^11 - 4825*x^10 + 34115*x^9 - 44010*x^8 + 177525*x^7 - 261205*x^6 + 418175*x^5 - 439550*x^4 + 133680*x^3 + 14000*x^2 - 215600*x + 54880, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 95 x^{13} - 270 x^{12} + 2665 x^{11} - 4825 x^{10} + 34115 x^{9} - 44010 x^{8} + 177525 x^{7} - 261205 x^{6} + 418175 x^{5} - 439550 x^{4} + 133680 x^{3} + 14000 x^{2} - 215600 x + 54880 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15330693091321490625000000000000=2^{12}\cdot 3^{13}\cdot 5^{17}\cdot 79^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $119.96$
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:  $2, 3, 5, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{18} a^{10} + \frac{1}{18} a^{8} - \frac{1}{9} a^{6} - \frac{1}{2} a^{5} - \frac{1}{18} a^{4} - \frac{1}{2} a^{3} - \frac{1}{18} a^{2} - \frac{1}{2} a + \frac{1}{9}$, $\frac{1}{18} a^{11} + \frac{1}{18} a^{9} + \frac{1}{18} a^{7} - \frac{1}{18} a^{5} - \frac{1}{18} a^{3} - \frac{1}{18} a$, $\frac{1}{36} a^{12} - \frac{1}{36} a^{11} - \frac{1}{36} a^{10} + \frac{1}{18} a^{9} - \frac{1}{36} a^{8} - \frac{1}{36} a^{7} - \frac{1}{12} a^{6} - \frac{2}{9} a^{5} - \frac{17}{36} a^{4} - \frac{11}{36} a^{3} - \frac{17}{36} a^{2} - \frac{2}{9} a - \frac{4}{9}$, $\frac{1}{3024} a^{13} - \frac{11}{1008} a^{12} - \frac{17}{3024} a^{11} - \frac{23}{1512} a^{10} + \frac{229}{3024} a^{9} - \frac{205}{3024} a^{8} + \frac{11}{3024} a^{7} + \frac{199}{1512} a^{6} - \frac{415}{3024} a^{5} - \frac{167}{432} a^{4} - \frac{481}{3024} a^{3} - \frac{451}{1512} a^{2} - \frac{109}{252} a + \frac{11}{54}$, $\frac{1}{577096279895821687434999967139424} a^{14} - \frac{5446118611244729575961157967}{192365426631940562478333322379808} a^{13} - \frac{6079973055188989323919159256645}{577096279895821687434999967139424} a^{12} + \frac{2849152559280726245252091031483}{288548139947910843717499983569712} a^{11} + \frac{14142639641852949119669432618821}{577096279895821687434999967139424} a^{10} + \frac{34541635642974958706506575167399}{577096279895821687434999967139424} a^{9} + \frac{4825177855298056681200514263599}{577096279895821687434999967139424} a^{8} + \frac{17300900632453120227854330600869}{288548139947910843717499983569712} a^{7} + \frac{63055114792518524439175312282625}{577096279895821687434999967139424} a^{6} + \frac{5878165615175761674787954205197}{82442325699403098204999995305632} a^{5} + \frac{13053875276625595723307328209243}{577096279895821687434999967139424} a^{4} - \frac{8948243224258100045754529252477}{288548139947910843717499983569712} a^{3} - \frac{2475750162099439021140281245853}{16030452219328380206527776864984} a^{2} + \frac{3239979334542133172046952722581}{10305290712425387275624999413204} a - \frac{12163377611568033976630077872}{40894010763592806649305553227}$

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:  $8$
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:  \( 22787447885.74553 \) (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

3.3.5925.1, 5.1.4050000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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}$
$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$$[5/4]_{12}^{2}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$