Normalized defining polynomial
\( x^{20} + 61 x^{18} + 1544 x^{16} + 21554 x^{14} + 184919 x^{12} + 1020980 x^{10} + 3666211 x^{8} + 8396334 x^{6} + 11588274 x^{4} + 8541234 x^{2} + 2518569 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(22923990352931577546297600000000000000=2^{20}\cdot 3^{6}\cdot 5^{14}\cdot 23^{8}\cdot 89^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $73.79$ | 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: | $2, 3, 5, 23, 89$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} - \frac{8}{23} a^{12} + \frac{3}{23} a^{10} + \frac{3}{23} a^{8} - \frac{1}{23} a^{6} + \frac{10}{23} a^{4} + \frac{11}{23} a^{2}$, $\frac{1}{23} a^{15} - \frac{8}{23} a^{13} + \frac{3}{23} a^{11} + \frac{3}{23} a^{9} - \frac{1}{23} a^{7} + \frac{10}{23} a^{5} + \frac{11}{23} a^{3}$, $\frac{1}{1587} a^{16} - \frac{8}{1587} a^{14} + \frac{509}{1587} a^{12} + \frac{716}{1587} a^{10} + \frac{620}{1587} a^{8} + \frac{608}{1587} a^{6} - \frac{449}{1587} a^{4} + \frac{5}{23} a^{2}$, $\frac{1}{1587} a^{17} - \frac{8}{1587} a^{15} + \frac{509}{1587} a^{13} + \frac{716}{1587} a^{11} + \frac{620}{1587} a^{9} + \frac{608}{1587} a^{7} - \frac{449}{1587} a^{5} + \frac{5}{23} a^{3}$, $\frac{1}{184366551} a^{18} + \frac{14012}{61455517} a^{16} + \frac{3628483}{184366551} a^{14} - \frac{30242262}{61455517} a^{12} - \frac{13321171}{61455517} a^{10} + \frac{8342272}{61455517} a^{8} + \frac{85195415}{184366551} a^{6} - \frac{1854653}{8015937} a^{4} - \frac{51947}{116173} a^{2} - \frac{822}{5051}$, $\frac{1}{184366551} a^{19} + \frac{14012}{61455517} a^{17} + \frac{3628483}{184366551} a^{15} - \frac{30242262}{61455517} a^{13} - \frac{13321171}{61455517} a^{11} + \frac{8342272}{61455517} a^{9} + \frac{85195415}{184366551} a^{7} - \frac{1854653}{8015937} a^{5} - \frac{51947}{116173} a^{3} - \frac{822}{5051} a$
Class group and class number
$C_{2}\times C_{2}\times C_{16896}$, which has order $67584$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 366014.001413 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 122880 |
| The 138 conjugacy class representatives for t20n794 are not computed |
| Character table for t20n794 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1 |
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 | R | R | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 | ||||||
| $3$ | 3.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 3.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 3.8.6.1 | $x^{8} + 9 x^{4} + 36$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ | |
| $5$ | 5.6.5.1 | $x^{6} - 5$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |
| 5.6.5.1 | $x^{6} - 5$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $23$ | 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 23.4.2.2 | $x^{4} - 23 x^{2} + 3703$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89 | Data not computed | ||||||