Normalized defining polynomial
\( x^{20} - 3 x^{19} + 3 x^{18} - 3 x^{17} + 8 x^{16} - 13 x^{15} + 16 x^{14} - 18 x^{13} + 22 x^{12} - 30 x^{11} + 32 x^{10} - 25 x^{9} + 18 x^{8} - 17 x^{7} + 25 x^{6} - 31 x^{5} + 29 x^{4} - 21 x^{3} + 11 x^{2} - 4 x + 1 \)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(2904089910121808158573\)\(\medspace = 67\cdot 83^{2}\cdot 631\cdot 1777^{4}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $11.83$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $67, 83, 631, 1777$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $2$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} - \frac{7}{17} a^{17} + \frac{5}{17} a^{16} + \frac{6}{17} a^{15} + \frac{7}{17} a^{14} + \frac{7}{17} a^{13} - \frac{7}{17} a^{12} - \frac{2}{17} a^{11} + \frac{8}{17} a^{10} + \frac{7}{17} a^{9} - \frac{3}{17} a^{7} - \frac{4}{17} a^{6} - \frac{8}{17} a^{5} + \frac{8}{17} a^{4} - \frac{8}{17} a^{3} + \frac{6}{17} a^{2} - \frac{7}{17} a + \frac{2}{17}$, $\frac{1}{188207} a^{19} - \frac{4904}{188207} a^{18} - \frac{634}{188207} a^{17} + \frac{84848}{188207} a^{16} - \frac{2209}{188207} a^{15} + \frac{3189}{11071} a^{14} + \frac{61458}{188207} a^{13} - \frac{63405}{188207} a^{12} + \frac{7099}{188207} a^{11} - \frac{84644}{188207} a^{10} - \frac{89733}{188207} a^{9} - \frac{47280}{188207} a^{8} + \frac{36481}{188207} a^{7} + \frac{25394}{188207} a^{6} - \frac{84355}{188207} a^{5} + \frac{54826}{188207} a^{4} - \frac{64382}{188207} a^{3} + \frac{34803}{188207} a^{2} + \frac{89973}{188207} a + \frac{253}{188207}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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 | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 194.247728439 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
| A non-solvable group of order 3932160 |
| The 506 conjugacy class representatives for t20n1015 are not computed |
| Character table for t20n1015 is not computed |
Intermediate fields
| 5.1.1777.1, 10.0.262091507.1 |
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 | $20$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | $16{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | $20$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 67 | Data not computed | ||||||
| $83$ | 83.4.0.1 | $x^{4} - x + 22$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 83.4.0.1 | $x^{4} - x + 22$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 83.4.0.1 | $x^{4} - x + 22$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 83.4.2.1 | $x^{4} + 249 x^{2} + 27556$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 83.4.0.1 | $x^{4} - x + 22$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 631 | Data not computed | ||||||
| 1777 | Data not computed | ||||||