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Properties

Label 15.15.3314662367...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 3^{20}\cdot 5^{6}\cdot 19^{2}\cdot 53^{4}\cdot 773^{4}\cdot 1510646329^{2}$
Root discriminant $7968.24$
Ramified primes $2, 3, 5, 19, 53, 773, 1510646329$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T98

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-167809024000, -880997376000, -1994829772800, -2541782310400, -1993571205120, -988894842240, -306744441216, -56522264736, -5385465792, -114496200, 20914272, 1275552, -16848, -2106, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2106*x^13 - 16848*x^12 + 1275552*x^11 + 20914272*x^10 - 114496200*x^9 - 5385465792*x^8 - 56522264736*x^7 - 306744441216*x^6 - 988894842240*x^5 - 1993571205120*x^4 - 2541782310400*x^3 - 1994829772800*x^2 - 880997376000*x - 167809024000)
 
gp: K = bnfinit(x^15 - 2106*x^13 - 16848*x^12 + 1275552*x^11 + 20914272*x^10 - 114496200*x^9 - 5385465792*x^8 - 56522264736*x^7 - 306744441216*x^6 - 988894842240*x^5 - 1993571205120*x^4 - 2541782310400*x^3 - 1994829772800*x^2 - 880997376000*x - 167809024000, 1)
 

Normalized defining polynomial

\( x^{15} - 2106 x^{13} - 16848 x^{12} + 1275552 x^{11} + 20914272 x^{10} - 114496200 x^{9} - 5385465792 x^{8} - 56522264736 x^{7} - 306744441216 x^{6} - 988894842240 x^{5} - 1993571205120 x^{4} - 2541782310400 x^{3} - 1994829772800 x^{2} - 880997376000 x - 167809024000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33146623676447473240676184720625960099206358036156416000000=2^{18}\cdot 3^{20}\cdot 5^{6}\cdot 19^{2}\cdot 53^{4}\cdot 773^{4}\cdot 1510646329^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $7968.24$
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:  $2, 3, 5, 19, 53, 773, 1510646329$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{6} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{80} a^{7} - \frac{1}{40} a^{5} + \frac{1}{20} a^{4} - \frac{1}{2} a$, $\frac{1}{800} a^{8} + \frac{3}{400} a^{6} + \frac{1}{100} a^{5} + \frac{2}{25} a^{4} - \frac{1}{50} a^{3} + \frac{7}{100} a^{2} - \frac{1}{5}$, $\frac{1}{3200} a^{9} + \frac{3}{1600} a^{7} - \frac{1}{100} a^{6} - \frac{1}{200} a^{5} + \frac{1}{50} a^{4} - \frac{93}{400} a^{3} - \frac{1}{10} a^{2} - \frac{1}{20} a$, $\frac{1}{121600} a^{10} - \frac{17}{60800} a^{8} - \frac{17}{7600} a^{7} + \frac{7}{3800} a^{6} - \frac{13}{3800} a^{5} + \frac{747}{15200} a^{4} - \frac{39}{380} a^{3} - \frac{71}{760} a^{2} - \frac{7}{38} a - \frac{1}{19}$, $\frac{1}{2432000} a^{11} + \frac{97}{1216000} a^{9} - \frac{93}{152000} a^{8} - \frac{1701}{304000} a^{7} - \frac{659}{76000} a^{6} - \frac{337}{60800} a^{5} + \frac{4707}{38000} a^{4} + \frac{17}{4750} a^{3} - \frac{4317}{19000} a^{2} - \frac{67}{3800} a - \frac{11}{50}$, $\frac{1}{24320000} a^{12} - \frac{43}{12160000} a^{10} + \frac{97}{1520000} a^{9} - \frac{1271}{3040000} a^{8} + \frac{4191}{760000} a^{7} - \frac{2209}{608000} a^{6} - \frac{2593}{380000} a^{5} - \frac{10049}{380000} a^{4} + \frac{15573}{190000} a^{3} - \frac{3229}{38000} a^{2} + \frac{151}{9500} a - \frac{41}{475}$, $\frac{1}{97280000000} a^{13} - \frac{31}{2432000000} a^{12} + \frac{4827}{48640000000} a^{11} + \frac{1681}{3040000000} a^{10} + \frac{51783}{1520000000} a^{9} - \frac{1205309}{3040000000} a^{8} - \frac{10232229}{2432000000} a^{7} + \frac{4392963}{380000000} a^{6} + \frac{41257967}{3040000000} a^{5} + \frac{10361443}{760000000} a^{4} - \frac{10474419}{152000000} a^{3} - \frac{2539109}{19000000} a^{2} + \frac{2515669}{7600000} a - \frac{361253}{1900000}$, $\frac{1}{12451840000000} a^{14} - \frac{11}{3112960000000} a^{13} + \frac{59307}{6225920000000} a^{12} - \frac{250003}{1556480000000} a^{11} + \frac{592771}{194560000000} a^{10} - \frac{378373}{389120000000} a^{9} - \frac{203591401}{1556480000000} a^{8} - \frac{96871929}{20480000000} a^{7} + \frac{3720417951}{389120000000} a^{6} + \frac{255139447}{12160000000} a^{5} - \frac{9817790267}{97280000000} a^{4} - \frac{953378699}{4864000000} a^{3} - \frac{606692383}{4864000000} a^{2} + \frac{22356689}{121600000} a + \frac{10827353}{60800000}$

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:  $14$
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:  \( 3920360683090000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T98:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 71 conjugacy class representatives for [1/2.S(5)^3]3 are not computed
Character table for [1/2.S(5)^3]3 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: 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 $15$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ R $15$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ $15$ $15$ $15$ R ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.2$x^{6} + 4 x^{2} - 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
53Data not computed
773Data not computed
1510646329Data not computed