Normalized defining polynomial
\( x^{20} - x^{19} + x^{18} + 2 x^{16} - 3 x^{15} - 3 x^{14} + 2 x^{13} + 4 x^{12} - 5 x^{10} + 4 x^{8} + 2 x^{7} - 3 x^{6} - 3 x^{5} + 2 x^{4} + x^{2} - x + 1 \)
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: | \(39479125871598264344535849=3^{10}\cdot 401^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $19.05$ | 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: | $3, 401$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{69} a^{18} + \frac{11}{23} a^{17} + \frac{6}{23} a^{16} + \frac{9}{23} a^{15} + \frac{5}{69} a^{14} + \frac{2}{69} a^{13} - \frac{3}{23} a^{12} - \frac{10}{23} a^{11} + \frac{28}{69} a^{10} + \frac{16}{69} a^{9} + \frac{28}{69} a^{8} - \frac{10}{23} a^{7} - \frac{3}{23} a^{6} + \frac{2}{69} a^{5} + \frac{5}{69} a^{4} + \frac{9}{23} a^{3} + \frac{6}{23} a^{2} + \frac{11}{23} a + \frac{1}{69}$, $\frac{1}{69} a^{19} + \frac{11}{23} a^{17} - \frac{5}{23} a^{16} + \frac{11}{69} a^{15} - \frac{25}{69} a^{14} - \frac{2}{23} a^{13} - \frac{3}{23} a^{12} - \frac{17}{69} a^{11} - \frac{11}{69} a^{10} - \frac{17}{69} a^{9} + \frac{4}{23} a^{8} + \frac{5}{23} a^{7} + \frac{1}{3} a^{6} + \frac{8}{69} a^{5} + \frac{8}{23} a^{3} - \frac{3}{23} a^{2} + \frac{16}{69} a - \frac{11}{23}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{2}{69} a^{19} - \frac{26}{69} a^{18} + \frac{14}{23} a^{17} - \frac{8}{23} a^{16} - \frac{34}{69} a^{15} - \frac{11}{69} a^{14} + \frac{98}{69} a^{13} + \frac{15}{23} a^{12} - \frac{221}{69} a^{11} - \frac{85}{69} a^{10} + \frac{170}{69} a^{9} + \frac{145}{69} a^{8} - \frac{49}{23} a^{7} - \frac{157}{69} a^{6} + \frac{70}{69} a^{5} + \frac{146}{69} a^{4} + \frac{3}{23} a^{3} - \frac{35}{23} a^{2} + \frac{7}{69} a + \frac{40}{69} \) (order $6$) | 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: | \( 45722.0230067 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_2^4:D_5$ (as 20T73):
| A solvable group of order 320 |
| The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$ |
| Character table for $C_2\times C_2^4:D_5$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 5.5.160801.1, 10.4.77570884803.1, 10.0.6283241669043.1, 10.6.2094413889681.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 401 | Data not computed | ||||||