Normalized defining polynomial
\( x^{20} - 10 x^{19} + 54 x^{18} - 201 x^{17} + 571 x^{16} - 1304 x^{15} + 2466 x^{14} - 3928 x^{13} + 5312 x^{12} - 6106 x^{11} + 5931 x^{10} - 4801 x^{9} + 3165 x^{8} - 1640 x^{7} + 655 x^{6} - 234 x^{5} + 136 x^{4} - 110 x^{3} + 47 x^{2} - 4 x - 1 \)
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: | \(1808152241408540960456444013=3^{9}\cdot 17^{12}\cdot 23\cdot 109^{2}\cdot 577\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.06$ | 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: | $3, 17, 23, 109, 577$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23371} a^{18} - \frac{9}{23371} a^{17} - \frac{10677}{23371} a^{16} - \frac{7864}{23371} a^{15} + \frac{343}{23371} a^{14} - \frac{5721}{23371} a^{13} - \frac{11654}{23371} a^{12} - \frac{524}{23371} a^{11} - \frac{5761}{23371} a^{10} - \frac{2579}{23371} a^{9} + \frac{3241}{23371} a^{8} + \frac{2585}{23371} a^{7} + \frac{8425}{23371} a^{6} + \frac{8421}{23371} a^{5} + \frac{5141}{23371} a^{4} - \frac{2882}{23371} a^{3} + \frac{7641}{23371} a^{2} - \frac{11498}{23371} a + \frac{473}{23371}$, $\frac{1}{70113} a^{19} + \frac{1}{70113} a^{18} - \frac{34138}{70113} a^{17} + \frac{2221}{70113} a^{16} - \frac{2728}{23371} a^{15} - \frac{2291}{70113} a^{14} + \frac{24620}{70113} a^{13} - \frac{7860}{23371} a^{12} - \frac{34372}{70113} a^{11} + \frac{3308}{23371} a^{10} + \frac{274}{23371} a^{9} + \frac{11624}{70113} a^{8} - \frac{12467}{70113} a^{7} - \frac{271}{23371} a^{6} - \frac{4133}{70113} a^{5} + \frac{25157}{70113} a^{4} + \frac{2192}{70113} a^{3} + \frac{18170}{70113} a^{2} + \frac{8573}{23371} a + \frac{4730}{70113}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 946949.013691 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 655360 |
| The 331 conjugacy class representatives for t20n946 are not computed |
| Character table for t20n946 is not computed |
Intermediate fields
| 5.1.44217.1, 10.2.213110596701.1 |
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 | $20$ | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ | $20$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ | R | $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ | $16{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ | $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | $20$ |
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 | Data not computed | ||||||
| $17$ | 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.8.6.1 | $x^{8} - 119 x^{4} + 23409$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 17.8.6.1 | $x^{8} - 119 x^{4} + 23409$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 23 | Data not computed | ||||||
| $109$ | 109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.8.0.1 | $x^{8} - x + 6$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 109.8.0.1 | $x^{8} - x + 6$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 577 | Data not computed | ||||||