Normalized defining polynomial
\( x^{20} - 2 x^{19} - 18 x^{18} + 80 x^{17} - 8 x^{16} - 528 x^{15} + 650 x^{14} + 1288 x^{13} - 2362 x^{12} - 2020 x^{11} + 5100 x^{10} + 2818 x^{9} - 7673 x^{8} - 4766 x^{7} + 6906 x^{6} + 5736 x^{5} - 2650 x^{4} - 2290 x^{3} + 320 x^{2} + 124 x - 17 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(9964694368301019319239964622848=2^{30}\cdot 13^{7}\cdot 101^{2}\cdot 347^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $35.48$ | 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: | $2, 13, 101, 347$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13} a^{18} + \frac{5}{13} a^{17} - \frac{5}{13} a^{16} - \frac{2}{13} a^{14} + \frac{4}{13} a^{13} - \frac{6}{13} a^{12} + \frac{1}{13} a^{11} - \frac{1}{13} a^{9} - \frac{3}{13} a^{8} - \frac{2}{13} a^{7} - \frac{3}{13} a^{6} + \frac{2}{13} a^{5} + \frac{5}{13} a^{4} - \frac{6}{13} a^{3} + \frac{6}{13} a^{2} + \frac{3}{13} a + \frac{1}{13}$, $\frac{1}{242187022338895147412591893362227} a^{19} + \frac{8092517108736230407880363439311}{242187022338895147412591893362227} a^{18} - \frac{58304318160364019157793012392983}{242187022338895147412591893362227} a^{17} - \frac{15639186719082898091191531419351}{242187022338895147412591893362227} a^{16} - \frac{115107263466968953346934243003530}{242187022338895147412591893362227} a^{15} - \frac{17949701984650522068552760961890}{242187022338895147412591893362227} a^{14} - \frac{8351209416940626437085250011705}{242187022338895147412591893362227} a^{13} + \frac{988853028002526697847372165512}{14246295431699714553681876080131} a^{12} - \frac{31040545518376424566364712954717}{242187022338895147412591893362227} a^{11} + \frac{46295995113298958164737472522909}{242187022338895147412591893362227} a^{10} - \frac{8129447485185460803949500584189}{242187022338895147412591893362227} a^{9} - \frac{53815503880965636951024913302505}{242187022338895147412591893362227} a^{8} - \frac{70614891398618543534466622431597}{242187022338895147412591893362227} a^{7} + \frac{106729804011693131937728126106046}{242187022338895147412591893362227} a^{6} + \frac{46911956115812950474273979775496}{242187022338895147412591893362227} a^{5} - \frac{89790524020350305382339197182592}{242187022338895147412591893362227} a^{4} + \frac{27595601594802937253133147422449}{242187022338895147412591893362227} a^{3} - \frac{16967203430741693374447030295147}{242187022338895147412591893362227} a^{2} - \frac{52628984276893686548810802524324}{242187022338895147412591893362227} a - \frac{6045368187634071702783260488913}{14246295431699714553681876080131}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 250518586.933 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 245760 |
| The 201 conjugacy class representatives for t20n886 are not computed |
| Character table for t20n886 is not computed |
Intermediate fields
| 5.3.4511.1, 10.6.2104587490304.1 |
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 | R | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ | $20$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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 | ||||||
| $13$ | $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $101$ | $\Q_{101}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{101}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{101}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{101}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 101.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 101.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 101.4.0.1 | $x^{4} - x + 12$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 101.4.0.1 | $x^{4} - x + 12$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 101.4.2.1 | $x^{4} + 505 x^{2} + 91809$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 347 | Data not computed | ||||||