Normalized defining polynomial
\( x^{18} + 93 x^{16} + 3519 x^{14} + 70211 x^{12} + 806046 x^{10} + 5497452 x^{8} + 22162597 x^{6} + 50450268 x^{4} + 57830979 x^{2} + 24303817 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-3192433564329392175935038228267008=-\,2^{18}\cdot 3^{24}\cdot 73^{3}\cdot 577^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $72.67$ | 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: | $2, 3, 73, 577$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{1338478772482475230368709} a^{16} - \frac{332647846955187838522400}{1338478772482475230368709} a^{14} - \frac{432315387063673240070172}{1338478772482475230368709} a^{12} - \frac{260333863097751915250948}{1338478772482475230368709} a^{10} - \frac{430826171568044152676690}{1338478772482475230368709} a^{8} - \frac{443930818011601899606732}{1338478772482475230368709} a^{6} + \frac{76561039361657464275199}{1338478772482475230368709} a^{4} - \frac{637494805538735934606037}{1338478772482475230368709} a^{2} + \frac{36058982740776751805}{2319720576226126915717}$, $\frac{1}{1338478772482475230368709} a^{17} - \frac{332647846955187838522400}{1338478772482475230368709} a^{15} - \frac{432315387063673240070172}{1338478772482475230368709} a^{13} - \frac{260333863097751915250948}{1338478772482475230368709} a^{11} - \frac{430826171568044152676690}{1338478772482475230368709} a^{9} - \frac{443930818011601899606732}{1338478772482475230368709} a^{7} + \frac{76561039361657464275199}{1338478772482475230368709} a^{5} - \frac{637494805538735934606037}{1338478772482475230368709} a^{3} + \frac{36058982740776751805}{2319720576226126915717} a$
Class group and class number
$C_{2}\times C_{2}\times C_{10388}$, which has order $41552$ (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: | \( 26510.9946997 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 82944 |
| The 110 conjugacy class representatives for t18n765 are not computed |
| Character table for t18n765 is not computed |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 9.9.22384826361.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 siblings: | 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 | $18$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ | $18$ | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | $18$ | $18$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$ |
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 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| $73$ | 73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 73.2.0.1 | $x^{2} - x + 11$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.4.2.1 | $x^{4} + 1533 x^{2} + 644809$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 73.4.0.1 | $x^{4} - x + 13$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 73.6.0.1 | $x^{6} - x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 577 | Data not computed | ||||||