Normalized defining polynomial
\( x^{18} - 6 x^{17} + 7 x^{16} + 25 x^{15} - 78 x^{14} + 53 x^{13} + 112 x^{12} - 204 x^{11} - 206 x^{10} + 631 x^{9} + 183 x^{8} - 1128 x^{7} + 281 x^{6} + 627 x^{5} - 124 x^{4} - 225 x^{3} + 65 x - 13 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[10, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2736002364807036533203125=3^{12}\cdot 5^{10}\cdot 13^{5}\cdot 17^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $22.78$ | 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: | $3, 5, 13, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{85} a^{15} - \frac{7}{17} a^{14} - \frac{29}{85} a^{13} - \frac{1}{5} a^{12} - \frac{16}{85} a^{11} - \frac{24}{85} a^{10} - \frac{1}{17} a^{9} - \frac{42}{85} a^{8} + \frac{2}{5} a^{7} - \frac{8}{85} a^{6} - \frac{8}{85} a^{5} + \frac{3}{17} a^{4} + \frac{28}{85} a^{3} - \frac{36}{85} a^{2} + \frac{27}{85} a - \frac{1}{85}$, $\frac{1}{85} a^{16} + \frac{21}{85} a^{14} - \frac{12}{85} a^{13} - \frac{16}{85} a^{12} + \frac{11}{85} a^{11} + \frac{1}{17} a^{10} + \frac{38}{85} a^{9} + \frac{9}{85} a^{8} - \frac{8}{85} a^{7} - \frac{33}{85} a^{6} - \frac{2}{17} a^{5} - \frac{42}{85} a^{4} + \frac{9}{85} a^{3} + \frac{42}{85} a^{2} + \frac{9}{85} a - \frac{7}{17}$, $\frac{1}{9501354655} a^{17} + \frac{3547965}{1900270931} a^{16} - \frac{4803965}{1900270931} a^{15} + \frac{2522359953}{9501354655} a^{14} + \frac{3678212583}{9501354655} a^{13} + \frac{4531506413}{9501354655} a^{12} - \frac{2347443819}{9501354655} a^{11} + \frac{75163334}{730873435} a^{10} - \frac{3447442451}{9501354655} a^{9} - \frac{1342552636}{9501354655} a^{8} + \frac{2863883888}{9501354655} a^{7} + \frac{3207483933}{9501354655} a^{6} - \frac{3839360354}{9501354655} a^{5} + \frac{205911808}{730873435} a^{4} - \frac{3661711086}{9501354655} a^{3} + \frac{23905107}{146174687} a^{2} - \frac{176083734}{730873435} a - \frac{487139}{42992555}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 445391.922114 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 92897280 |
| The 168 conjugacy class representatives for t18n966 are not computed |
| Character table for t18n966 is not computed |
Intermediate fields
| 9.5.22253180625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $18$ | R | R | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ | R | R | $18$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | $18$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 3.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.12.12.12 | $x^{12} + 165 x^{10} - 312 x^{9} - 288 x^{8} - 180 x^{7} - 36 x^{6} - 135 x^{5} - 243 x^{4} + 54 x^{3} + 81 x^{2} + 81 x - 162$ | $3$ | $4$ | $12$ | 12T41 | $[3/2, 3/2]_{2}^{4}$ | |
| $5$ | 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.12.10.4 | $x^{12} - 5 x^{6} + 50$ | $6$ | $2$ | $10$ | $C_3\times (C_3 : C_4)$ | $[\ ]_{6}^{6}$ | |
| $13$ | 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.3.2.1 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.3.2.1 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.10.0.1 | $x^{10} + 2 x^{2} - 2 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| $17$ | 17.6.5.2 | $x^{6} + 51$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |
| 17.12.0.1 | $x^{12} + 3 x^{2} - 2 x + 5$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |