Normalized defining polynomial
\( x^{15} - 1044 x^{13} - 16704 x^{12} + 133652 x^{11} + 6281344 x^{10} + 89085240 x^{9} + 1072877184 x^{8} + 14132201344 x^{7} + 140824572928 x^{6} + 896700835840 x^{5} + 3610996899840 x^{4} + 9207966105600 x^{3} + 14453106278400 x^{2} + 12766150656000 x + 4863295488000 \)
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: | \(634826879298290317710532545900028146989740835908935880704000000=2^{18}\cdot 3^{6}\cdot 5^{6}\cdot 41^{2}\cdot 61^{10}\cdot 773^{4}\cdot 4261^{2}\cdot 165379^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15{,}376.01$ | 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: | $2, 3, 5, 41, 61, 773, 4261, 165379$ | 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 \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} + \frac{1}{10} a^{3} + \frac{1}{10} a^{2}$, $\frac{1}{40} a^{6} + \frac{1}{20} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{160} a^{7} - \frac{1}{40} a^{5} - \frac{1}{20} a^{4} - \frac{3}{40} a^{3} - \frac{1}{5} a^{2} - \frac{1}{4} a$, $\frac{1}{3200} a^{8} - \frac{9}{800} a^{6} + \frac{1}{50} a^{5} - \frac{99}{800} a^{4} + \frac{6}{25} a^{3} + \frac{23}{400} a^{2} + \frac{1}{5}$, $\frac{1}{102400} a^{9} - \frac{29}{25600} a^{7} - \frac{9}{1600} a^{6} - \frac{219}{25600} a^{5} + \frac{1}{800} a^{4} - \frac{3017}{12800} a^{3} - \frac{31}{160} a^{2} - \frac{39}{160} a - \frac{1}{4}$, $\frac{1}{1228800} a^{10} + \frac{1}{102400} a^{8} - \frac{3}{6400} a^{7} + \frac{1189}{307200} a^{6} - \frac{43}{1920} a^{5} + \frac{6269}{51200} a^{4} + \frac{71}{3200} a^{3} - \frac{57}{3200} a^{2} - \frac{7}{16} a + \frac{2}{5}$, $\frac{1}{196608000} a^{11} - \frac{1}{4915200} a^{10} + \frac{53}{16384000} a^{9} - \frac{139}{2048000} a^{8} - \frac{72587}{49152000} a^{7} + \frac{3467}{6144000} a^{6} - \frac{106493}{4915200} a^{5} + \frac{55357}{1024000} a^{4} + \frac{10493}{256000} a^{3} - \frac{341}{4000} a^{2} + \frac{1049}{6400} a + \frac{363}{800}$, $\frac{1}{3932160000} a^{12} - \frac{241}{983040000} a^{10} + \frac{63}{20480000} a^{9} + \frac{39733}{983040000} a^{8} - \frac{89867}{30720000} a^{7} - \frac{176057}{19660800} a^{6} - \frac{449407}{30720000} a^{5} - \frac{582817}{5120000} a^{4} + \frac{121417}{640000} a^{3} - \frac{5927}{128000} a^{2} - \frac{999}{2000} a + \frac{187}{400}$, $\frac{1}{7864320000000} a^{13} - \frac{1}{16384000000} a^{12} + \frac{1499}{1966080000000} a^{11} + \frac{5823}{40960000000} a^{10} - \frac{4145147}{1966080000000} a^{9} + \frac{4156679}{30720000000} a^{8} - \frac{1977473}{39321600000} a^{7} - \frac{694281277}{61440000000} a^{6} - \frac{17200739}{20480000000} a^{5} - \frac{123827631}{2560000000} a^{4} + \frac{41598223}{256000000} a^{3} - \frac{1107361}{8000000} a^{2} + \frac{556427}{3200000} a + \frac{25901}{400000}$, $\frac{1}{8053063680000000} a^{14} - \frac{43}{1006632960000000} a^{13} - \frac{77607}{671088640000000} a^{12} + \frac{131807}{83886080000000} a^{11} - \frac{67460201}{671088640000000} a^{10} + \frac{198343311}{83886080000000} a^{9} - \frac{112588117817}{1006632960000000} a^{8} + \frac{90418189807}{41943040000000} a^{7} - \frac{361740633389}{62914560000000} a^{6} + \frac{20463474667}{1966080000000} a^{5} + \frac{106182460207}{1310720000000} a^{4} - \frac{2763934753}{32768000000} a^{3} + \frac{2338189443}{16384000000} a^{2} + \frac{10764871}{40960000} a + \frac{1349067}{51200000}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | \( 133364352976000000000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| 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
| 3.3.3721.1 |
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 | ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | $15$ | $15$ | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ | ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ | R | ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ | $15$ | ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.6.9.1 | $x^{6} + 4 x^{4} + 4 x^{2} - 8$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
| 2.6.9.1 | $x^{6} + 4 x^{4} + 4 x^{2} - 8$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
| $3$ | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.5.4.1 | $x^{5} - 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| $5$ | 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 | $x^{6} - 25 x^{2} + 250$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 41 | Data not computed | ||||||
| 61 | Data not computed | ||||||
| 773 | Data not computed | ||||||
| 4261 | Data not computed | ||||||
| 165379 | Data not computed | ||||||