Normalized defining polynomial
\( x^{18} + 2 x^{16} - 73 x^{14} - 206 x^{12} + 1152 x^{10} + 3126 x^{8} - 6344 x^{6} - 14675 x^{4} + 10000 x^{2} + 15625 \)
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: | \(-17621893739736239772433439801344=-\,2^{14}\cdot 32009^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $54.44$ | 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: | $2, 32009$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{150} a^{14} - \frac{8}{25} a^{12} + \frac{26}{75} a^{10} + \frac{22}{75} a^{8} + \frac{26}{75} a^{6} + \frac{13}{75} a^{4} + \frac{28}{75} a^{2} + \frac{1}{6}$, $\frac{1}{1500} a^{15} - \frac{1}{300} a^{14} - \frac{4}{125} a^{13} + \frac{4}{25} a^{12} + \frac{101}{750} a^{11} + \frac{49}{150} a^{10} + \frac{97}{750} a^{9} + \frac{53}{150} a^{8} + \frac{101}{750} a^{7} + \frac{49}{150} a^{6} - \frac{181}{375} a^{5} + \frac{31}{75} a^{4} - \frac{197}{750} a^{3} + \frac{47}{150} a^{2} + \frac{1}{60} a - \frac{1}{12}$, $\frac{1}{1140363810000} a^{16} - \frac{2996452073}{1140363810000} a^{14} - \frac{64406159299}{570181905000} a^{12} + \frac{13299356129}{31676772500} a^{10} + \frac{10747149857}{31676772500} a^{8} + \frac{652507099}{6553815000} a^{6} - \frac{212943850147}{570181905000} a^{4} - \frac{101831479}{3040970160} a^{2} - \frac{580156133}{1824582096}$, $\frac{1}{11403638100000} a^{17} - \frac{1}{2280727620000} a^{16} - \frac{2996452073}{11403638100000} a^{15} + \frac{2996452073}{2280727620000} a^{14} - \frac{64406159299}{5701819050000} a^{13} + \frac{64406159299}{1140363810000} a^{12} + \frac{108329673629}{316767725000} a^{11} + \frac{18377416371}{63353545000} a^{10} - \frac{52606395143}{316767725000} a^{9} - \frac{10747149857}{63353545000} a^{8} - \frac{19008937901}{65538150000} a^{7} + \frac{5901307901}{13107630000} a^{6} + \frac{357238054853}{5701819050000} a^{5} - \frac{357238054853}{1140363810000} a^{4} + \frac{9021079001}{30409701600} a^{3} - \frac{2939138681}{6081940320} a^{2} - \frac{6053902421}{18245820960} a - \frac{1244425963}{3649164192}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 2217810618.94 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 6912 |
| The 48 conjugacy class representatives for t18n521 |
| Character table for t18n521 is not computed |
Intermediate fields
| 3.3.32009.3, 9.9.32795655776729.1 |
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 | ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.6.6.3 | $x^{6} + 2 x^{4} + x^{2} - 7$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.4 | $x^{6} + x^{2} + 1$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ | |
| 32009 | Data not computed | ||||||