Normalized defining polynomial
\( x^{20} - 4 x^{19} - 2 x^{18} + 30 x^{17} - 29 x^{16} - 88 x^{15} + 222 x^{14} - 70 x^{13} - 398 x^{12} + 668 x^{11} - 300 x^{10} - 382 x^{9} + 684 x^{8} - 460 x^{7} + 166 x^{6} - 78 x^{5} + 29 x^{4} + 44 x^{3} - 30 x^{2} - 8 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1734361690558178085830656=2^{20}\cdot 31^{4}\cdot 53^{4}\cdot 61^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.29$ | 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: | $2, 31, 53, 61$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{2}$, $\frac{1}{185194843566022} a^{19} + \frac{12348793414427}{92597421783011} a^{18} + \frac{6341366782737}{185194843566022} a^{17} - \frac{3197726395630}{92597421783011} a^{16} - \frac{10404035143345}{185194843566022} a^{15} - \frac{3231420875877}{92597421783011} a^{14} + \frac{6872293653985}{92597421783011} a^{13} - \frac{44399380508475}{185194843566022} a^{12} + \frac{32136954252377}{185194843566022} a^{11} - \frac{3367216385706}{92597421783011} a^{10} - \frac{41586326367184}{92597421783011} a^{9} - \frac{31967871345701}{185194843566022} a^{8} + \frac{6205420146185}{185194843566022} a^{7} - \frac{10014361249614}{92597421783011} a^{6} + \frac{17188483898017}{92597421783011} a^{5} + \frac{3222813239717}{185194843566022} a^{4} + \frac{36254852113157}{92597421783011} a^{3} - \frac{34349527093018}{92597421783011} a^{2} + \frac{10063812901813}{185194843566022} a - \frac{37668645660255}{185194843566022}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 29938.8300885 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 122880 |
| The 252 conjugacy class representatives for t20n791 are not computed |
| Character table for t20n791 is not computed |
Intermediate fields
| 5.3.26288.1, 10.6.42154595584.1 |
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 | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.8.0.1}{8} }$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ | $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ | $20$ | $16{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ | ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.8.0.1}{8} }$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$ |
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 | Data not computed | ||||||
| $31$ | 31.8.4.1 | $x^{8} + 32674 x^{4} - 119164 x^{2} + 266897569$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 31.12.0.1 | $x^{12} + 2 x^{2} - 3 x + 22$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $53$ | 53.8.4.1 | $x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 53.12.0.1 | $x^{12} - x + 12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| 61 | Data not computed | ||||||