Normalized defining polynomial
\( x^{15} - 1386 x^{13} - 11088 x^{12} + 616072 x^{11} + 10189792 x^{10} - 36042840 x^{9} - 2150494272 x^{8} - 22831399776 x^{7} - 124100912256 x^{6} - 400169331840 x^{5} - 806742097920 x^{4} - 1028587686400 x^{3} - 807251404800 x^{2} - 356514816000 x - 67907584000 \)
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[5, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-63587282469726653922176018224241933346768105472000000=-\,2^{18}\cdot 5^{6}\cdot 23^{5}\cdot 59^{4}\cdot 149\cdot 281^{4}\cdot 9133^{2}\cdot 50683^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $3313.02$ | 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: | $2, 5, 23, 59, 149, 281, 9133, 50683$ | 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 \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{6} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{80} a^{7} - \frac{1}{40} a^{5} + \frac{1}{20} a^{4} - \frac{1}{2} a$, $\frac{1}{800} a^{8} + \frac{3}{400} a^{6} + \frac{1}{100} a^{5} + \frac{2}{25} a^{4} - \frac{1}{50} a^{3} + \frac{7}{100} a^{2} - \frac{1}{5}$, $\frac{1}{3200} a^{9} + \frac{3}{1600} a^{7} - \frac{1}{100} a^{6} + \frac{3}{400} a^{5} + \frac{1}{50} a^{4} + \frac{97}{400} a^{3} + \frac{1}{5} a^{2} + \frac{9}{20} a$, $\frac{1}{6400} a^{10} - \frac{1}{3200} a^{8} - \frac{1}{200} a^{7} - \frac{3}{800} a^{6} + \frac{33}{800} a^{4} - \frac{13}{100} a^{3} + \frac{31}{200} a^{2} + \frac{1}{5}$, $\frac{1}{128000} a^{11} - \frac{3}{64000} a^{9} - \frac{3}{8000} a^{8} - \frac{13}{8000} a^{7} + \frac{11}{4000} a^{6} + \frac{41}{3200} a^{5} + \frac{97}{2000} a^{4} - \frac{299}{2000} a^{3} - \frac{137}{1000} a^{2} - \frac{67}{200} a + \frac{1}{50}$, $\frac{1}{1280000} a^{12} + \frac{17}{640000} a^{10} + \frac{7}{80000} a^{9} + \frac{11}{40000} a^{8} - \frac{99}{40000} a^{7} - \frac{67}{6400} a^{6} + \frac{37}{20000} a^{5} - \frac{1067}{10000} a^{4} + \frac{1393}{10000} a^{3} + \frac{291}{2000} a^{2} - \frac{209}{500} a + \frac{9}{25}$, $\frac{1}{5120000000} a^{13} - \frac{31}{128000000} a^{12} + \frac{5187}{2560000000} a^{11} - \frac{1039}{160000000} a^{10} + \frac{97429}{640000000} a^{9} - \frac{86799}{160000000} a^{8} - \frac{129907}{25600000} a^{7} - \frac{51173}{5000000} a^{6} + \frac{3168897}{160000000} a^{5} - \frac{2263887}{40000000} a^{4} + \frac{399471}{8000000} a^{3} - \frac{246319}{1000000} a^{2} - \frac{25721}{400000} a + \frac{35177}{100000}$, $\frac{1}{655360000000} a^{14} - \frac{11}{163840000000} a^{13} + \frac{59667}{327680000000} a^{12} - \frac{253243}{81920000000} a^{11} + \frac{2692853}{81920000000} a^{10} + \frac{102059}{2560000000} a^{9} + \frac{17953909}{81920000000} a^{8} + \frac{109899639}{20480000000} a^{7} + \frac{77225841}{20480000000} a^{6} + \frac{10497079}{1280000000} a^{5} - \frac{191075497}{5120000000} a^{4} - \frac{5738009}{256000000} a^{3} - \frac{33905653}{256000000} a^{2} + \frac{1259099}{6400000} a - \frac{1200077}{3200000}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 977308953441000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 10368000 |
| The 140 conjugacy class representatives for [S(5)^3]S(3)=S(5)wrS(3) are not computed |
| Character table for [S(5)^3]S(3)=S(5)wrS(3) is not computed |
Intermediate fields
| 3.1.23.1 |
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 | ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ | ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ | ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | R | $15$ | $15$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ | R |
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 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.6.9.6 | $x^{6} + 4 x^{2} + 8$ | $2$ | $3$ | $9$ | $A_4\times C_2$ | $[2, 2, 3]^{3}$ | |
| 2.6.9.8 | $x^{6} + 4 x^{2} - 24$ | $2$ | $3$ | $9$ | $A_4\times C_2$ | $[2, 2, 3]^{3}$ | |
| $5$ | $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.2 | $x^{4} - 5 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| $23$ | $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.6.3.2 | $x^{6} - 529 x^{2} + 48668$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 59 | Data not computed | ||||||
| 149 | Data not computed | ||||||
| 281 | Data not computed | ||||||
| 9133 | Data not computed | ||||||
| 50683 | Data not computed | ||||||