Normalized defining polynomial
\( x^{18} - 33 x^{16} - 11 x^{15} + 387 x^{14} + 144 x^{13} - 2216 x^{12} - 623 x^{11} + 6888 x^{10} + 821 x^{9} - 11951 x^{8} + 859 x^{7} + 11003 x^{6} - 2794 x^{5} - 4431 x^{4} + 1723 x^{3} + 391 x^{2} - 150 x - 9 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[18, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(28195670721946845353167339457=7^{8}\cdot 257^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $38.07$ | 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: | $7, 257$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{13} + \frac{3}{7} a^{12} - \frac{3}{7} a^{11} + \frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{2} + \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{15} + \frac{2}{7} a^{13} + \frac{1}{7} a^{12} - \frac{3}{7} a^{11} + \frac{2}{7} a^{10} - \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{301} a^{16} + \frac{3}{43} a^{15} + \frac{20}{301} a^{14} - \frac{128}{301} a^{13} + \frac{44}{301} a^{12} - \frac{94}{301} a^{11} + \frac{109}{301} a^{10} + \frac{40}{301} a^{9} + \frac{94}{301} a^{8} + \frac{132}{301} a^{7} + \frac{15}{301} a^{6} - \frac{33}{301} a^{5} - \frac{68}{301} a^{4} - \frac{61}{301} a^{3} + \frac{118}{301} a^{2} + \frac{13}{43} a - \frac{138}{301}$, $\frac{1}{2053892695071} a^{17} + \frac{217865605}{684630898357} a^{16} - \frac{44513065399}{684630898357} a^{15} + \frac{15690548032}{2053892695071} a^{14} - \frac{246893784840}{684630898357} a^{13} + \frac{177326458737}{684630898357} a^{12} - \frac{1286523230}{47764946397} a^{11} - \frac{596734620842}{2053892695071} a^{10} - \frac{42089872179}{97804414051} a^{9} - \frac{371828267662}{2053892695071} a^{8} + \frac{377392395004}{2053892695071} a^{7} + \frac{45332509624}{2053892695071} a^{6} - \frac{760230506080}{2053892695071} a^{5} + \frac{1001610200534}{2053892695071} a^{4} + \frac{43217687647}{97804414051} a^{3} + \frac{929280140935}{2053892695071} a^{2} - \frac{3209035691}{47764946397} a + \frac{157971768179}{684630898357}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $17$ | 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: | \( 290535844.764 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 54 |
| The 10 conjugacy class representatives for $D_9:C_3$ |
| Character table for $D_9:C_3$ |
Intermediate fields
| \(\Q(\sqrt{257}) \), 3.3.257.1 x3, 6.6.16974593.1, 9.9.10474291432801.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 9 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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ | R | ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 257 | Data not computed | ||||||