Normalized defining polynomial
\( x^{18} - 6 x^{17} + 16 x^{16} - 28 x^{15} + 22 x^{14} + 68 x^{13} - 346 x^{12} + 820 x^{11} - 1248 x^{10} + 1348 x^{9} - 984 x^{8} + 200 x^{7} + 1040 x^{6} - 904 x^{5} + 680 x^{4} - 136 x^{3} - 968 x^{2} - 8 x + 344 \)
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: | \(2945640892745755719979237376=2^{18}\cdot 37^{6}\cdot 16361^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $33.58$ | 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: | $2, 37, 16361$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{4} a^{14}$, $\frac{1}{4} a^{15}$, $\frac{1}{44} a^{16} - \frac{3}{44} a^{15} + \frac{1}{22} a^{14} + \frac{1}{11} a^{13} + \frac{1}{22} a^{12} + \frac{5}{22} a^{11} + \frac{1}{11} a^{10} - \frac{5}{22} a^{9} - \frac{5}{22} a^{7} - \frac{1}{22} a^{6} - \frac{5}{11} a^{5} - \frac{3}{11} a^{3} - \frac{4}{11} a^{2} + \frac{2}{11} a + \frac{4}{11}$, $\frac{1}{15175035435879869152196} a^{17} + \frac{28560546868262504362}{3793758858969967288049} a^{16} - \frac{930765061052370648065}{7587517717939934576098} a^{15} + \frac{1119222607005516272623}{15175035435879869152196} a^{14} + \frac{1868164926795137991423}{15175035435879869152196} a^{13} + \frac{1842865421216680162155}{15175035435879869152196} a^{12} + \frac{3513246061999043863}{689774337994539506918} a^{11} - \frac{1448381994006416489897}{7587517717939934576098} a^{10} - \frac{1684433864745497285935}{7587517717939934576098} a^{9} - \frac{1741103802055034898717}{7587517717939934576098} a^{8} - \frac{1811965413644063797063}{7587517717939934576098} a^{7} + \frac{1547294807582753666797}{7587517717939934576098} a^{6} + \frac{1783962850126807931374}{3793758858969967288049} a^{5} - \frac{896053689644980012578}{3793758858969967288049} a^{4} - \frac{804598906259004363952}{3793758858969967288049} a^{3} - \frac{304920114637698839678}{3793758858969967288049} a^{2} + \frac{867598245196799352050}{3793758858969967288049} a + \frac{1101391512856946803959}{3793758858969967288049}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 5265558.78179 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 331776 |
| The 165 conjugacy class representatives for t18n885 are not computed |
| Character table for t18n885 is not computed |
Intermediate fields
| 3.3.148.1, 9.9.53038958912.1 |
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 | R | ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $37$ | 37.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 37.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 37.4.2.1 | $x^{4} + 333 x^{2} + 34225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.8.4.1 | $x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 16361 | Data not computed | ||||||