Normalized defining polynomial
\( x^{18} - 153 x^{15} + 9991 x^{12} - 349506 x^{9} + 6810103 x^{6} - 68511168 x^{3} + 275894451 \)
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: | \(-242791248552429691086166766209415374323=-\,3^{27}\cdot 19^{6}\cdot 937^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $135.68$ | 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: | $3, 19, 937$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{19} a^{12} - \frac{9}{19} a^{9} + \frac{5}{19} a^{6} + \frac{3}{19} a^{3} + \frac{7}{19}$, $\frac{1}{19} a^{13} - \frac{9}{19} a^{10} + \frac{5}{19} a^{7} + \frac{3}{19} a^{4} + \frac{7}{19} a$, $\frac{1}{57} a^{14} - \frac{3}{19} a^{11} - \frac{14}{57} a^{8} + \frac{1}{19} a^{5} + \frac{7}{57} a^{2}$, $\frac{1}{7522279197303} a^{15} + \frac{35514901857}{2507426399101} a^{12} - \frac{177087405842}{7522279197303} a^{9} + \frac{1151216448055}{2507426399101} a^{6} - \frac{3737308996484}{7522279197303} a^{3} + \frac{1155418013284}{2507426399101}$, $\frac{1}{4897003757444253} a^{16} - \frac{8962108811953}{544111528604917} a^{13} - \frac{1146334891415957}{4897003757444253} a^{10} - \frac{250930770272824}{544111528604917} a^{7} - \frac{1216407897488015}{4897003757444253} a^{4} - \frac{103387121635559}{544111528604917} a$, $\frac{1}{1062649815365402901} a^{17} + \frac{1204523975143690}{354216605121800967} a^{14} - \frac{83364450608506310}{1062649815365402901} a^{11} + \frac{44036177728028380}{354216605121800967} a^{8} - \frac{274159933115038748}{1062649815365402901} a^{5} + \frac{78500097936184459}{354216605121800967} a^{2}$
Class group and class number
$C_{3}\times C_{3}\times C_{117}$, which has order $1053$ (assuming GRH)
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{14294274502}{55928937650810679} a^{17} - \frac{11947919291285}{354216605121800967} a^{14} + \frac{2009876394408790}{1062649815365402901} a^{11} - \frac{19475994463765748}{354216605121800967} a^{8} + \frac{871447114341310207}{1062649815365402901} a^{5} - \frac{1680573346045090745}{354216605121800967} a^{2} \) (order $18$) | 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: | \( 19430507563.475685 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3^2\times S_3$ (as 18T17):
| A solvable group of order 54 |
| The 27 conjugacy class representatives for $C_3^2\times S_3$ |
| Character table for $C_3^2\times S_3$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), 6.0.6238464041547.1, \(\Q(\zeta_{9})\), Deg 6, 6.0.8557563843.1 |
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 | ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
| 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
| 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
| $19$ | $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 937 | Data not computed | ||||||