Normalized defining polynomial
\( x^{20} - 36 x^{18} + 454 x^{16} - 2744 x^{14} + 8921 x^{12} - 16548 x^{10} + 17842 x^{8} - 10976 x^{6} + 3632 x^{4} - 576 x^{2} + 32 \)
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: | \(9676847870747939759856242687914016768=2^{59}\cdot 11^{8}\cdot 23^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $70.68$ | 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: | $2, 11, 23$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{92} a^{14} + \frac{1}{46} a^{12} + \frac{9}{46} a^{10} + \frac{2}{23} a^{8} + \frac{29}{92} a^{6} - \frac{21}{46} a^{4} + \frac{3}{46} a^{2} - \frac{10}{23}$, $\frac{1}{92} a^{15} + \frac{1}{46} a^{13} + \frac{9}{46} a^{11} + \frac{2}{23} a^{9} + \frac{29}{92} a^{7} - \frac{21}{46} a^{5} + \frac{3}{46} a^{3} - \frac{10}{23} a$, $\frac{1}{2024} a^{16} + \frac{5}{1012} a^{14} + \frac{247}{1012} a^{12} - \frac{169}{506} a^{10} + \frac{553}{2024} a^{8} + \frac{325}{1012} a^{6} - \frac{211}{1012} a^{4} + \frac{163}{506} a^{2} - \frac{86}{253}$, $\frac{1}{4048} a^{17} - \frac{3}{1012} a^{15} + \frac{225}{2024} a^{13} + \frac{119}{506} a^{11} - \frac{1647}{4048} a^{9} - \frac{503}{1012} a^{7} - \frac{761}{2024} a^{5} - \frac{94}{253} a^{3} - \frac{229}{506} a$, $\frac{1}{4048} a^{18} - \frac{1}{2024} a^{14} - \frac{21}{253} a^{12} + \frac{185}{4048} a^{10} + \frac{3}{253} a^{8} + \frac{917}{2024} a^{6} - \frac{95}{506} a^{4} - \frac{93}{253} a^{2} - \frac{98}{253}$, $\frac{1}{8096} a^{19} + \frac{21}{4048} a^{15} + \frac{111}{506} a^{13} - \frac{3071}{8096} a^{11} - \frac{114}{253} a^{9} - \frac{469}{4048} a^{7} - \frac{73}{1012} a^{5} + \frac{353}{1012} a^{3} - \frac{104}{253} a$
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: | \( 7502898268910 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_4\times A_5$ (as 20T63):
| A non-solvable group of order 240 |
| The 20 conjugacy class representatives for $C_4\times A_5$ |
| Character table for $C_4\times A_5$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 5.5.16386304.1, 10.10.2148087670243328.1 |
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 | $20$ | ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | R | $20$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ | $20$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.11.1 | $x^{4} + 12 x^{2} + 2$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ |
| 2.8.24.9 | $x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $[2, 3, 4]$ | |
| 2.8.24.9 | $x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $[2, 3, 4]$ | |
| $11$ | 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.12.8.1 | $x^{12} - 33 x^{9} + 363 x^{6} - 1331 x^{3} + 117128$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
| $23$ | 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.6.4.1 | $x^{6} + 460 x^{3} + 181447$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 23.6.4.1 | $x^{6} + 460 x^{3} + 181447$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |