Normalized defining polynomial
\( x^{20} + 2 x^{18} - 46 x^{16} - 44 x^{14} + 155 x^{12} + 1077 x^{10} + 2021 x^{8} + 2306 x^{6} + 2494 x^{4} + 1604 x^{2} + 401 \)
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: | \(148715055972301824959913581871104=2^{20}\cdot 23^{2}\cdot 401^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $40.61$ | 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: | $2, 23, 401$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{711} a^{16} - \frac{32}{237} a^{14} - \frac{347}{711} a^{12} - \frac{223}{711} a^{10} + \frac{91}{237} a^{8} + \frac{31}{711} a^{6} - \frac{85}{237} a^{4} + \frac{52}{711} a^{2} + \frac{335}{711}$, $\frac{1}{711} a^{17} - \frac{32}{237} a^{15} - \frac{347}{711} a^{13} - \frac{223}{711} a^{11} + \frac{91}{237} a^{9} + \frac{31}{711} a^{7} - \frac{85}{237} a^{5} + \frac{52}{711} a^{3} + \frac{335}{711} a$, $\frac{1}{10755155197449} a^{18} - \frac{5805107435}{10755155197449} a^{16} - \frac{290906610635}{10755155197449} a^{14} - \frac{12938955519}{398339081387} a^{12} + \frac{4724166873494}{10755155197449} a^{10} + \frac{4212434689009}{10755155197449} a^{8} - \frac{58502729252}{136141205031} a^{6} + \frac{2680401554518}{10755155197449} a^{4} + \frac{967858732216}{3585051732483} a^{2} - \frac{3316886161795}{10755155197449}$, $\frac{1}{10755155197449} a^{19} - \frac{5805107435}{10755155197449} a^{17} - \frac{290906610635}{10755155197449} a^{15} - \frac{12938955519}{398339081387} a^{13} + \frac{4724166873494}{10755155197449} a^{11} + \frac{4212434689009}{10755155197449} a^{9} - \frac{58502729252}{136141205031} a^{7} + \frac{2680401554518}{10755155197449} a^{5} + \frac{967858732216}{3585051732483} a^{3} - \frac{3316886161795}{10755155197449} a$
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: | \( 260153094.465 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 163840 |
| The 277 conjugacy class representatives for t20n848 are not computed |
| Character table for t20n848 is not computed |
Intermediate fields
| 5.5.160801.1, 10.4.594710116823.1 |
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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.10.4 | $x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2]^{10}$ |
| 2.10.10.6 | $x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2]^{10}$ | |
| $23$ | 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.8.0.1 | $x^{8} + x^{2} - 2 x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 401 | Data not computed | ||||||