Normalized defining polynomial
\( x^{20} - 5 x^{18} + 10 x^{16} - 20 x^{14} + 35 x^{12} - 75 x^{10} + 145 x^{8} - 190 x^{6} + 140 x^{4} - 50 x^{2} + 5 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1462323200000000000000000000000=2^{32}\cdot 5^{23}\cdot 13^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $32.23$ | 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: | $2, 5, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{14} + \frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{15} + \frac{1}{4} a^{13} + \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{456248} a^{18} + \frac{5937}{57031} a^{16} - \frac{1}{2} a^{15} - \frac{38229}{114062} a^{14} - \frac{40357}{228124} a^{12} - \frac{1}{2} a^{11} + \frac{198451}{456248} a^{10} - \frac{16557}{228124} a^{8} - \frac{32989}{456248} a^{6} + \frac{15477}{35096} a^{4} - \frac{1}{2} a^{3} - \frac{120201}{456248} a^{2} - \frac{1}{2} a + \frac{47845}{456248}$, $\frac{1}{456248} a^{19} + \frac{5937}{57031} a^{17} - \frac{38229}{114062} a^{15} - \frac{1}{2} a^{14} - \frac{40357}{228124} a^{13} + \frac{198451}{456248} a^{11} - \frac{1}{2} a^{10} - \frac{16557}{228124} a^{9} - \frac{32989}{456248} a^{7} + \frac{15477}{35096} a^{5} - \frac{120201}{456248} a^{3} - \frac{1}{2} a^{2} + \frac{47845}{456248} a - \frac{1}{2}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 19214814.0455 \) (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 t20n790 are not computed |
| Character table for t20n790 is not computed |
Intermediate fields
| 5.3.162500.1, 10.2.2112500000000.1 |
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 | ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ | $20$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.20.53 | $x^{8} + 4 x^{5} + 2 x^{4} + 2$ | $8$ | $1$ | $20$ | $Q_8:C_2$ | $[2, 3, 3]^{2}$ |
| 2.12.12.12 | $x^{12} + 66 x^{10} - 93 x^{8} - 68 x^{6} - 41 x^{4} + 66 x^{2} - 123$ | $2$ | $6$ | $12$ | 12T29 | $[2, 2, 2]^{6}$ | |
| 5 | Data not computed | ||||||
| $13$ | $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |