Normalized defining polynomial
\( x^{18} + 10 x^{16} - 119 x^{12} - 72 x^{10} + 214 x^{8} + 31 x^{6} - 68 x^{4} - 8 x^{2} + 1 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-6973355056864235107287040000=-\,2^{18}\cdot 5^{4}\cdot 37^{8}\cdot 59^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $35.23$ | 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: | $2, 5, 37, 59$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{28} a^{14} - \frac{1}{28} a^{12} - \frac{5}{14} a^{8} - \frac{1}{2} a^{7} - \frac{1}{7} a^{6} - \frac{1}{2} a^{5} - \frac{5}{14} a^{4} - \frac{1}{2} a^{3} - \frac{11}{28} a^{2} - \frac{1}{2} a + \frac{9}{28}$, $\frac{1}{28} a^{15} - \frac{1}{28} a^{13} - \frac{5}{14} a^{9} - \frac{1}{2} a^{8} - \frac{1}{7} a^{7} - \frac{1}{2} a^{6} - \frac{5}{14} a^{5} - \frac{1}{2} a^{4} - \frac{11}{28} a^{3} - \frac{1}{2} a^{2} + \frac{9}{28} a$, $\frac{1}{2137772} a^{16} + \frac{2801}{2137772} a^{14} - \frac{99527}{1068886} a^{12} + \frac{397665}{1068886} a^{10} - \frac{1}{2} a^{9} - \frac{77566}{534443} a^{8} - \frac{1}{2} a^{7} - \frac{74475}{1068886} a^{6} - \frac{1}{2} a^{5} - \frac{991651}{2137772} a^{4} - \frac{1}{2} a^{3} + \frac{716731}{2137772} a^{2} + \frac{522475}{1068886}$, $\frac{1}{2137772} a^{17} + \frac{2801}{2137772} a^{15} - \frac{99527}{1068886} a^{13} + \frac{397665}{1068886} a^{11} - \frac{1}{2} a^{10} - \frac{77566}{534443} a^{9} - \frac{1}{2} a^{8} - \frac{74475}{1068886} a^{7} - \frac{1}{2} a^{6} - \frac{991651}{2137772} a^{5} - \frac{1}{2} a^{4} + \frac{716731}{2137772} a^{3} + \frac{522475}{1068886} a$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $12$ | 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: | \( 62591436.5352 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 82944 |
| The 144 conjugacy class representatives for t18n772 are not computed |
| Character table for t18n772 is not computed |
Intermediate fields
| 3.3.148.1, 9.9.10438327105600.1 |
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 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $18$ | R | ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ | $18$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ | R | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | 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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.10.5 | $x^{6} + 2 x^{5} + 6$ | $6$ | $1$ | $10$ | $S_4\times C_2$ | $[2, 8/3, 8/3]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| $5$ | 5.3.2.1 | $x^{3} - 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 5.3.2.1 | $x^{3} - 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 5.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 5.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 37 | Data not computed | ||||||
| 59 | Data not computed | ||||||