Normalized defining polynomial
\( x^{18} - 4 x^{17} - 6 x^{16} + 48 x^{15} - 22 x^{14} - 306 x^{13} + 687 x^{12} + 346 x^{11} - 3471 x^{10} + 4492 x^{9} + 2094 x^{8} - 12089 x^{7} + 13476 x^{6} - 2955 x^{5} - 8918 x^{4} + 11417 x^{3} - 6158 x^{2} + 1494 x - 127 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[10, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(279992823820843547402730217=7^{12}\cdot 53^{6}\cdot 97^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $29.46$ | 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: | $7, 53, 97$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{97} a^{15} - \frac{13}{97} a^{14} - \frac{45}{97} a^{13} + \frac{36}{97} a^{12} + \frac{47}{97} a^{11} - \frac{13}{97} a^{10} + \frac{27}{97} a^{9} + \frac{27}{97} a^{8} - \frac{3}{97} a^{7} - \frac{39}{97} a^{6} + \frac{45}{97} a^{5} - \frac{45}{97} a^{4} - \frac{22}{97} a^{3} - \frac{32}{97} a^{2} - \frac{30}{97} a + \frac{47}{97}$, $\frac{1}{291} a^{16} + \frac{77}{291} a^{14} - \frac{64}{291} a^{13} - \frac{67}{291} a^{12} - \frac{27}{97} a^{11} + \frac{52}{291} a^{10} - \frac{10}{291} a^{9} - \frac{137}{291} a^{8} - \frac{26}{97} a^{7} + \frac{23}{291} a^{6} - \frac{14}{97} a^{5} - \frac{122}{291} a^{4} + \frac{70}{291} a^{3} + \frac{136}{291} a^{2} - \frac{52}{291} a + \frac{29}{291}$, $\frac{1}{54601763514567439347} a^{17} + \frac{85506809138736211}{54601763514567439347} a^{16} - \frac{222130858906824760}{54601763514567439347} a^{15} - \frac{25787385474108176594}{54601763514567439347} a^{14} + \frac{11862698351373614992}{54601763514567439347} a^{13} + \frac{2656882363064089538}{54601763514567439347} a^{12} + \frac{3968017424079211858}{54601763514567439347} a^{11} + \frac{6443071693989155162}{18200587838189146449} a^{10} + \frac{1122615579430295909}{18200587838189146449} a^{9} + \frac{6745814467057338862}{54601763514567439347} a^{8} + \frac{9670751826455812652}{54601763514567439347} a^{7} + \frac{10817932317672321596}{54601763514567439347} a^{6} + \frac{17958554459908737655}{54601763514567439347} a^{5} - \frac{2190096629799094369}{54601763514567439347} a^{4} - \frac{15067826164353338482}{54601763514567439347} a^{3} + \frac{1545888952750528654}{18200587838189146449} a^{2} + \frac{15359990934083084401}{54601763514567439347} a - \frac{16060268168203611454}{54601763514567439347}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 3343187.00358 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 2304 |
| The 48 conjugacy class representatives for t18n367 |
| Character table for t18n367 is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 3.3.2597.1, 9.9.17515230173.1 |
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 | ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ | R | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$ |
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 | Data not computed | ||||||
| $53$ | 53.3.0.1 | $x^{3} - x + 8$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 53.3.0.1 | $x^{3} - x + 8$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 53.12.6.1 | $x^{12} + 2382032 x^{6} - 418195493 x^{2} + 1418519112256$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| 97 | Data not computed | ||||||