Normalized defining polynomial
\( x^{20} - 10 x^{19} + 52 x^{18} - 183 x^{17} + 479 x^{16} - 976 x^{15} + 1492 x^{14} - 1422 x^{13} - 392 x^{12} + 5160 x^{11} - 12784 x^{10} + 20728 x^{9} - 25522 x^{8} + 25308 x^{7} - 21158 x^{6} + 15263 x^{5} - 9312 x^{4} + 4500 x^{3} - 1560 x^{2} + 336 x - 32 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(485822685091899819825439453125=3^{4}\cdot 5^{13}\cdot 23^{8}\cdot 89^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $30.50$ | 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: | $3, 5, 23, 89$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{40} a^{16} + \frac{1}{20} a^{15} - \frac{3}{40} a^{14} - \frac{9}{40} a^{13} + \frac{1}{5} a^{12} + \frac{3}{8} a^{11} + \frac{2}{5} a^{10} - \frac{7}{40} a^{9} + \frac{3}{10} a^{8} - \frac{3}{8} a^{7} + \frac{2}{5} a^{6} - \frac{7}{40} a^{5} + \frac{9}{20} a^{4} + \frac{1}{8} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a + \frac{1}{5}$, $\frac{1}{40} a^{17} + \frac{3}{40} a^{15} - \frac{3}{40} a^{14} - \frac{1}{10} a^{13} + \frac{9}{40} a^{12} + \frac{3}{20} a^{11} - \frac{9}{40} a^{10} + \frac{3}{20} a^{9} + \frac{11}{40} a^{8} - \frac{7}{20} a^{7} + \frac{11}{40} a^{6} + \frac{3}{10} a^{5} + \frac{19}{40} a^{4} - \frac{1}{20} a^{3} + \frac{7}{20} a^{2} - \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{3110250800} a^{18} - \frac{9}{3110250800} a^{17} + \frac{31773209}{3110250800} a^{16} - \frac{63546367}{777562700} a^{15} - \frac{24650879}{622050160} a^{14} + \frac{645649541}{3110250800} a^{13} - \frac{386929437}{3110250800} a^{12} - \frac{530924153}{3110250800} a^{11} - \frac{1449080487}{3110250800} a^{10} - \frac{328473}{124410032} a^{9} + \frac{1270577949}{3110250800} a^{8} + \frac{643509227}{3110250800} a^{7} + \frac{691198639}{3110250800} a^{6} + \frac{73833129}{3110250800} a^{5} + \frac{236782129}{622050160} a^{4} - \frac{108010989}{1555125400} a^{3} + \frac{75429169}{155512540} a^{2} + \frac{1863927}{194390675} a + \frac{6671922}{194390675}$, $\frac{1}{1570676654000} a^{19} + \frac{243}{1570676654000} a^{18} - \frac{11165131939}{1570676654000} a^{17} - \frac{1487441137}{157067665400} a^{16} - \frac{165572168411}{1570676654000} a^{15} - \frac{30025676649}{1570676654000} a^{14} + \frac{31297006929}{314135330800} a^{13} - \frac{151999993657}{1570676654000} a^{12} + \frac{506325016727}{1570676654000} a^{11} + \frac{203221839151}{1570676654000} a^{10} - \frac{553412612841}{1570676654000} a^{9} + \frac{60340221991}{314135330800} a^{8} + \frac{310670193113}{1570676654000} a^{7} - \frac{135991627143}{1570676654000} a^{6} - \frac{136577999817}{1570676654000} a^{5} + \frac{93702266041}{785338327000} a^{4} - \frac{164365222383}{785338327000} a^{3} - \frac{99614613297}{392669163500} a^{2} - \frac{8494623753}{196334581750} a + \frac{2147861964}{98167290875}$
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: | \( 15059790.9299 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 122880 |
| The 138 conjugacy class representatives for t20n804 are not computed |
| Character table for t20n804 is not computed |
Intermediate fields
| 5.5.767625.1, 10.6.311712266390625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ | R | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 3.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.6.5.2 | $x^{6} + 10$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ | |
| 5.6.4.1 | $x^{6} + 25 x^{3} + 200$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| $23$ | 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 23.8.4.1 | $x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 23.8.4.1 | $x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $89$ | 89.4.2.1 | $x^{4} + 979 x^{2} + 285156$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 89.4.2.1 | $x^{4} + 979 x^{2} + 285156$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.6.0.1 | $x^{6} - x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 89.6.0.1 | $x^{6} - x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |