Normalized defining polynomial
\( x^{20} - 2 x^{19} + 6 x^{18} - 3 x^{17} + 11 x^{16} - 4 x^{15} + 32 x^{14} - 18 x^{13} + 52 x^{12} - 27 x^{11} + 77 x^{10} - 50 x^{9} + 70 x^{8} - 44 x^{7} + 53 x^{6} - 24 x^{5} + 30 x^{4} - 27 x^{3} + 7 x^{2} - 2 x + 1 \)
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: | \(54320086096828525390625=5^{10}\cdot 13^{4}\cdot 41^{7}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.70$ | 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: | $5, 13, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7} a^{18} + \frac{1}{7} a^{17} - \frac{3}{7} a^{16} - \frac{3}{7} a^{15} + \frac{3}{7} a^{14} - \frac{1}{7} a^{13} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{8} + \frac{3}{7} a^{6} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{3}{7}$, $\frac{1}{954153880759121} a^{19} + \frac{7328833107142}{136307697251303} a^{18} - \frac{272982321462960}{954153880759121} a^{17} - \frac{29256956833333}{136307697251303} a^{16} - \frac{326558077876423}{954153880759121} a^{15} + \frac{391599725471776}{954153880759121} a^{14} - \frac{40844251575112}{954153880759121} a^{13} - \frac{125098059064068}{954153880759121} a^{12} - \frac{414317166761815}{954153880759121} a^{11} - \frac{346860234631364}{954153880759121} a^{10} - \frac{95970386665596}{954153880759121} a^{9} + \frac{227804907959064}{954153880759121} a^{8} + \frac{404632233072847}{954153880759121} a^{7} - \frac{341030604756845}{954153880759121} a^{6} - \frac{212469584531054}{954153880759121} a^{5} - \frac{438794628078468}{954153880759121} a^{4} + \frac{206428529135940}{954153880759121} a^{3} + \frac{86229213312181}{954153880759121} a^{2} + \frac{429421687076686}{954153880759121} a + \frac{30242605278344}{954153880759121}$
Class group and class number
Trivial group, which has order $1$
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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 852.798782372 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 960 |
| The 35 conjugacy class representatives for t20n174 |
| Character table for t20n174 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.0.1025.1, 5.1.2665.1, 10.2.887778125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | $20$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ | R | $20$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.6.3.1 | $x^{6} - 10 x^{4} + 25 x^{2} - 500$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.1 | $x^{6} - 10 x^{4} + 25 x^{2} - 500$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $13$ | 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 13.12.0.1 | $x^{12} + x^{2} - x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $41$ | 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |