Normalized defining polynomial
\( x^{20} - x^{19} - 35 x^{18} + 48 x^{17} + 371 x^{16} - 720 x^{15} - 1072 x^{14} + 3390 x^{13} - 1550 x^{12} - 4400 x^{11} + 11484 x^{10} - 4579 x^{9} - 21055 x^{8} + 17107 x^{7} + 19669 x^{6} - 16970 x^{5} - 10688 x^{4} + 7275 x^{3} + 3345 x^{2} - 1150 x - 475 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[16, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(133572012199627601368564605712890625=5^{18}\cdot 11^{6}\cdot 71^{4}\cdot 167^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $57.05$ | 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: | $5, 11, 71, 167$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} + \frac{1}{11} a^{15} + \frac{4}{11} a^{14} - \frac{1}{11} a^{12} - \frac{3}{11} a^{11} + \frac{4}{11} a^{10} + \frac{3}{11} a^{9} + \frac{3}{11} a^{8} + \frac{3}{11} a^{7} + \frac{1}{11} a^{4} - \frac{1}{11} a^{3} + \frac{3}{11} a^{2} + \frac{1}{11}$, $\frac{1}{11} a^{17} + \frac{3}{11} a^{15} - \frac{4}{11} a^{14} - \frac{1}{11} a^{13} - \frac{2}{11} a^{12} - \frac{4}{11} a^{11} - \frac{1}{11} a^{10} - \frac{3}{11} a^{7} + \frac{1}{11} a^{5} - \frac{2}{11} a^{4} + \frac{4}{11} a^{3} - \frac{3}{11} a^{2} + \frac{1}{11} a - \frac{1}{11}$, $\frac{1}{110} a^{18} + \frac{2}{55} a^{17} + \frac{19}{55} a^{15} - \frac{29}{110} a^{14} + \frac{1}{22} a^{13} - \frac{21}{55} a^{12} + \frac{5}{22} a^{11} - \frac{1}{22} a^{10} - \frac{2}{11} a^{9} - \frac{1}{110} a^{8} - \frac{27}{55} a^{7} - \frac{1}{11} a^{6} - \frac{53}{110} a^{5} + \frac{2}{55} a^{4} + \frac{1}{22} a^{3} - \frac{53}{110} a^{2} + \frac{5}{22} a + \frac{3}{22}$, $\frac{1}{16326662525241707281610} a^{19} - \frac{6134928396869997308}{8163331262620853640805} a^{18} + \frac{18894543338554686890}{1632666252524170728161} a^{17} + \frac{34200538657786693499}{8163331262620853640805} a^{16} - \frac{3055777247975876153369}{16326662525241707281610} a^{15} + \frac{993607672893583882329}{3265332505048341456322} a^{14} + \frac{117716005315498423274}{8163331262620853640805} a^{13} + \frac{634880890110352650275}{3265332505048341456322} a^{12} - \frac{1051778414853791310273}{3265332505048341456322} a^{11} - \frac{636133695588992880258}{1632666252524170728161} a^{10} + \frac{5977527354858235010339}{16326662525241707281610} a^{9} - \frac{3975536769177002146157}{8163331262620853640805} a^{8} - \frac{495697412132988511895}{1632666252524170728161} a^{7} + \frac{6382669640040857608727}{16326662525241707281610} a^{6} + \frac{519174287726997918502}{8163331262620853640805} a^{5} - \frac{1563905963779003275983}{3265332505048341456322} a^{4} + \frac{3911730412412695357157}{16326662525241707281610} a^{3} - \frac{595574177390247531827}{3265332505048341456322} a^{2} + \frac{68772362465016207699}{296848409549849223302} a + \frac{519475344197654128373}{1632666252524170728161}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $17$ | 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: | \( 169345636339 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 928972800 |
| The 139 conjugacy class representatives for t20n1100 are not computed |
| Character table for t20n1100 is not computed |
Intermediate fields
| 10.10.6645000909765625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ | R | $18{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | $18{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ | $18{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.2.2 | $x^{4} - 5 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 5.5.8.7 | $x^{5} + 10 x^{4} + 5$ | $5$ | $1$ | $8$ | $F_5$ | $[2]^{4}$ | |
| 5.5.8.7 | $x^{5} + 10 x^{4} + 5$ | $5$ | $1$ | $8$ | $F_5$ | $[2]^{4}$ | |
| 5.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $11$ | 11.8.6.3 | $x^{8} - 11 x^{4} + 847$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ |
| 11.12.0.1 | $x^{12} - x + 7$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $71$ | 71.2.0.1 | $x^{2} - x + 11$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 71.2.0.1 | $x^{2} - x + 11$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.4.2.1 | $x^{4} + 1491 x^{2} + 609961$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 71.4.2.1 | $x^{4} + 1491 x^{2} + 609961$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 71.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 71.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $167$ | $\Q_{167}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{167}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 167.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 167.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 167.3.0.1 | $x^{3} - x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 167.3.0.1 | $x^{3} - x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 167.4.2.2 | $x^{4} - 167 x^{2} + 139445$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 167.4.2.2 | $x^{4} - 167 x^{2} + 139445$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |