Normalized defining polynomial
\( x^{22} - 11 x^{21} + 34 x^{20} + 45 x^{19} - 419 x^{18} + 294 x^{17} + 2039 x^{16} - 2950 x^{15} - 5716 x^{14} + 11278 x^{13} + 10840 x^{12} - 25208 x^{11} - 15904 x^{10} + 35866 x^{9} + 19724 x^{8} - 31917 x^{7} - 19447 x^{6} + 15438 x^{5} + 12389 x^{4} - 1977 x^{3} - 3512 x^{2} - 887 x - 59 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[14, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1252214624578349298453638315804341=113\cdot 29739173\cdot 610429790897^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $31.95$ | 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: | $113, 29739173, 610429790897$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{97} a^{20} - \frac{10}{97} a^{19} - \frac{13}{97} a^{18} + \frac{14}{97} a^{17} - \frac{21}{97} a^{16} + \frac{46}{97} a^{15} - \frac{48}{97} a^{14} - \frac{44}{97} a^{13} - \frac{7}{97} a^{12} - \frac{2}{97} a^{11} + \frac{39}{97} a^{10} + \frac{28}{97} a^{9} + \frac{44}{97} a^{8} - \frac{46}{97} a^{7} + \frac{8}{97} a^{6} - \frac{40}{97} a^{5} + \frac{5}{97} a^{4} + \frac{45}{97} a^{3} + \frac{27}{97} a^{2} - \frac{26}{97} a + \frac{22}{97}$, $\frac{1}{97} a^{21} - \frac{16}{97} a^{19} - \frac{19}{97} a^{18} + \frac{22}{97} a^{17} + \frac{30}{97} a^{16} + \frac{24}{97} a^{15} - \frac{39}{97} a^{14} + \frac{38}{97} a^{13} + \frac{25}{97} a^{12} + \frac{19}{97} a^{11} + \frac{30}{97} a^{10} + \frac{33}{97} a^{9} + \frac{6}{97} a^{8} + \frac{33}{97} a^{7} + \frac{40}{97} a^{6} - \frac{7}{97} a^{5} - \frac{2}{97} a^{4} - \frac{8}{97} a^{3} - \frac{47}{97} a^{2} - \frac{44}{97} a + \frac{26}{97}$
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: | \( 443688211.731 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 81749606400 |
| The 752 conjugacy class representatives for t22n53 are not computed |
| Character table for t22n53 is not computed |
Intermediate fields
| 11.7.610429790897.1 |
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 | $22$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | $18{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ | $16{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ | $22$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ | $22$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | $22$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ | $20{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $113$ | $\Q_{113}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{113}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 113.2.1.2 | $x^{2} + 339$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 113.4.0.1 | $x^{4} - x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 113.14.0.1 | $x^{14} - x + 93$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
| 29739173 | Data not computed | ||||||
| 610429790897 | Data not computed | ||||||