Normalized defining polynomial
\( x^{18} - 3 x^{17} - 6 x^{16} + 47 x^{15} - 63 x^{14} + 59 x^{13} + 392 x^{12} - 1311 x^{11} + 959 x^{10} + 2245 x^{9} - 5596 x^{8} + 2424 x^{7} + 5370 x^{6} - 7713 x^{5} + 1764 x^{4} + 3911 x^{3} - 2988 x^{2} + 404 x + 1 \)
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: | \(-18105575687315621060510012299=-\,19^{3}\cdot 1129^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $37.14$ | 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: | $19, 1129$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{16958629589923219461007634886337} a^{17} + \frac{732087483244326590298003278151}{1541693599083929041909784989667} a^{16} - \frac{3346359150267046510469105398270}{16958629589923219461007634886337} a^{15} - \frac{1679081079160872237375130263684}{16958629589923219461007634886337} a^{14} - \frac{1160616048273867968274566681616}{16958629589923219461007634886337} a^{13} + \frac{1548594931989169000972109893928}{16958629589923219461007634886337} a^{12} + \frac{8349023592849808435499765237519}{16958629589923219461007634886337} a^{11} - \frac{768884631745656066757320095112}{16958629589923219461007634886337} a^{10} + \frac{4820003312050272276690750988172}{16958629589923219461007634886337} a^{9} + \frac{5537718578902318749429463257732}{16958629589923219461007634886337} a^{8} - \frac{1794689402264784966105620017214}{16958629589923219461007634886337} a^{7} + \frac{7979149169045643027470923340433}{16958629589923219461007634886337} a^{6} + \frac{2084876349548589409830731271292}{16958629589923219461007634886337} a^{5} + \frac{2660478569592504908890250375947}{16958629589923219461007634886337} a^{4} + \frac{104375001169837687759364753027}{997566446466071733000449110961} a^{3} - \frac{5144985684331307883867479921657}{16958629589923219461007634886337} a^{2} + \frac{23269426760274726462414813140}{997566446466071733000449110961} a + \frac{7066507613131871281963075280853}{16958629589923219461007634886337}$
Class group and class number
Trivial group, which has order $1$
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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 17188110.2671 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 2304 |
| The 40 conjugacy class representatives for t18n375 |
| Character table for t18n375 is not computed |
Intermediate fields
| 3.3.1129.1, 9.9.1624709678881.1 |
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 | $18$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | R | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $19$ | 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.4.2.2 | $x^{4} - 19 x^{2} + 722$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 1129 | Data not computed | ||||||