Normalized defining polynomial
\( x^{20} - 40 x^{18} + 680 x^{16} - 2 x^{15} - 6400 x^{14} + 60 x^{13} + 36400 x^{12} - 720 x^{11} - 128126 x^{10} + 4400 x^{9} + 274520 x^{8} - 14400 x^{7} - 337640 x^{6} + 24252 x^{5} + 210400 x^{4} - 18520 x^{3} - 50400 x^{2} + 5040 x - 124 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[20, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(28964050250000000000000000000000000000=2^{28}\cdot 5^{30}\cdot 41^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $74.66$ | 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: | $2, 5, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{10} a^{10} + \frac{2}{5} a^{5} + \frac{2}{5}$, $\frac{1}{10} a^{11} + \frac{2}{5} a^{6} + \frac{2}{5} a$, $\frac{1}{10} a^{12} + \frac{2}{5} a^{7} + \frac{2}{5} a^{2}$, $\frac{1}{10} a^{13} + \frac{2}{5} a^{8} + \frac{2}{5} a^{3}$, $\frac{1}{10} a^{14} + \frac{2}{5} a^{9} + \frac{2}{5} a^{4}$, $\frac{1}{50} a^{15} + \frac{1}{50} a^{10} + \frac{11}{25} a^{5} - \frac{1}{25}$, $\frac{1}{50} a^{16} + \frac{1}{50} a^{11} + \frac{11}{25} a^{6} - \frac{1}{25} a$, $\frac{1}{50} a^{17} + \frac{1}{50} a^{12} + \frac{11}{25} a^{7} - \frac{1}{25} a^{2}$, $\frac{1}{1550} a^{18} - \frac{4}{775} a^{17} - \frac{1}{310} a^{16} - \frac{7}{1550} a^{15} + \frac{3}{62} a^{14} + \frac{33}{775} a^{13} + \frac{21}{775} a^{12} - \frac{2}{155} a^{11} + \frac{33}{1550} a^{10} + \frac{13}{31} a^{9} - \frac{259}{775} a^{8} - \frac{38}{775} a^{7} - \frac{32}{155} a^{6} + \frac{328}{775} a^{5} + \frac{4}{31} a^{4} - \frac{46}{775} a^{3} + \frac{58}{775} a^{2} - \frac{7}{31} a + \frac{2}{25}$, $\frac{1}{1550} a^{19} - \frac{7}{1550} a^{17} + \frac{3}{310} a^{16} - \frac{6}{775} a^{15} + \frac{23}{775} a^{14} - \frac{1}{31} a^{13} + \frac{34}{775} a^{12} - \frac{13}{310} a^{11} - \frac{47}{1550} a^{10} + \frac{326}{775} a^{9} - \frac{10}{31} a^{8} + \frac{373}{775} a^{7} - \frac{54}{155} a^{6} - \frac{252}{775} a^{5} + \frac{289}{775} a^{4} + \frac{382}{775} a^{2} + \frac{6}{31} a + \frac{7}{25}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $19$ | 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: | \( 4532487073970 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$D_5^2:C_4$ (as 20T94):
| A solvable group of order 400 |
| The 28 conjugacy class representatives for $D_5^2:C_4$ |
| Character table for $D_5^2:C_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.16400.1, 10.10.13132812500000000.1 |
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 | $20$ | R | $20$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | $20$ | $20$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$ |
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 | ||||||
| $5$ | 5.10.15.5 | $x^{10} - 15 x^{6} - 20 x^{5} + 5$ | $10$ | $1$ | $15$ | $C_5^2 : C_4$ | $[5/4, 7/4]_{4}$ |
| 5.10.15.5 | $x^{10} - 15 x^{6} - 20 x^{5} + 5$ | $10$ | $1$ | $15$ | $C_5^2 : C_4$ | $[5/4, 7/4]_{4}$ | |
| $41$ | 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |