Normalized defining polynomial
\( x^{18} - 5 x^{17} + 11 x^{16} - 29 x^{15} + 53 x^{14} - 42 x^{13} + 22 x^{12} - 36 x^{11} + 135 x^{10} - 158 x^{9} + 183 x^{8} + 102 x^{7} + 25 x^{6} - 187 x^{5} + 100 x^{4} + 107 x^{3} - 166 x^{2} + 54 x - 1 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[6, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(161020573979625730382141821=61\cdot 1129^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $28.57$ | 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: | $61, 1129$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{703} a^{16} + \frac{281}{703} a^{15} - \frac{140}{703} a^{14} + \frac{77}{703} a^{13} + \frac{3}{37} a^{12} + \frac{39}{703} a^{11} + \frac{346}{703} a^{10} - \frac{65}{703} a^{9} + \frac{128}{703} a^{8} - \frac{336}{703} a^{7} + \frac{202}{703} a^{6} - \frac{312}{703} a^{5} + \frac{248}{703} a^{4} + \frac{40}{703} a^{3} + \frac{88}{703} a^{2} - \frac{270}{703} a - \frac{15}{703}$, $\frac{1}{111825382654906018619} a^{17} + \frac{49217157229136144}{111825382654906018619} a^{16} - \frac{48998816960922699467}{111825382654906018619} a^{15} + \frac{1764429231724196714}{5885546455521369401} a^{14} - \frac{49809746236939261500}{111825382654906018619} a^{13} + \frac{33489268984711205786}{111825382654906018619} a^{12} + \frac{3575983911782068620}{111825382654906018619} a^{11} - \frac{8868592852557313585}{111825382654906018619} a^{10} + \frac{37938584149468079476}{111825382654906018619} a^{9} + \frac{10035090107793521332}{111825382654906018619} a^{8} - \frac{2209655492802735225}{111825382654906018619} a^{7} - \frac{11288351458694490421}{111825382654906018619} a^{6} + \frac{46078887102684393333}{111825382654906018619} a^{5} - \frac{21307414408733501667}{111825382654906018619} a^{4} + \frac{2089196655107581448}{111825382654906018619} a^{3} - \frac{42663007161196758429}{111825382654906018619} a^{2} + \frac{51572867481655131538}{111825382654906018619} a - \frac{38780250988298178536}{111825382654906018619}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $11$ | 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: | \( 1698918.3841 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 9216 |
| The 88 conjugacy class representatives for t18n548 are not computed |
| Character table for t18n548 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 | $18$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ | $18$ | ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ | $18$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$ |
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 |
|---|---|---|---|---|---|---|---|
| $61$ | 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 61.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 61.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 1129 | Data not computed | ||||||