Normalized defining polynomial
\( x^{20} - 3 x^{19} - 4 x^{18} + 18 x^{17} + 9 x^{16} - 52 x^{15} - 27 x^{14} + 97 x^{13} + 71 x^{12} - 99 x^{11} - 131 x^{10} + 44 x^{9} + 108 x^{8} + 17 x^{7} - 14 x^{6} - 36 x^{5} - 18 x^{4} + 16 x^{3} + 10 x^{2} - 7 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(3919569348575304934929089=11^{18}\cdot 89^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.97$ | 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: | $11, 89$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{89} a^{18} - \frac{26}{89} a^{16} + \frac{29}{89} a^{15} - \frac{44}{89} a^{14} - \frac{21}{89} a^{13} - \frac{12}{89} a^{12} - \frac{11}{89} a^{11} + \frac{35}{89} a^{10} - \frac{19}{89} a^{9} + \frac{21}{89} a^{8} - \frac{9}{89} a^{7} - \frac{25}{89} a^{6} - \frac{38}{89} a^{5} - \frac{23}{89} a^{4} + \frac{19}{89} a^{3} + \frac{11}{89} a^{2} - \frac{13}{89} a - \frac{4}{89}$, $\frac{1}{26375599361} a^{19} + \frac{131295033}{26375599361} a^{18} + \frac{259733302}{26375599361} a^{17} - \frac{4942931396}{26375599361} a^{16} + \frac{5119255268}{26375599361} a^{15} + \frac{7885539713}{26375599361} a^{14} - \frac{11514834064}{26375599361} a^{13} - \frac{8229992977}{26375599361} a^{12} - \frac{12356080093}{26375599361} a^{11} - \frac{9594466828}{26375599361} a^{10} + \frac{4371952209}{26375599361} a^{9} + \frac{11317242384}{26375599361} a^{8} + \frac{3642178053}{26375599361} a^{7} - \frac{1051730326}{26375599361} a^{6} - \frac{2533481625}{26375599361} a^{5} - \frac{11760270951}{26375599361} a^{4} + \frac{11577821394}{26375599361} a^{3} + \frac{7135154283}{26375599361} a^{2} - \frac{8899532947}{26375599361} a + \frac{7264262560}{26375599361}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 41392.1603807 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 10240 |
| The 136 conjugacy class representatives for t20n432 are not computed |
| Character table for t20n432 is not computed |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.8.209857344499.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | $20$ | $20$ | ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
| $11$ | 11.10.9.7 | $x^{10} + 2673$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.9.7 | $x^{10} + 2673$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
| $89$ | $\Q_{89}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{89}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{89}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{89}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{89}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{89}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{89}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{89}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 89.4.0.1 | $x^{4} - x + 27$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.0.1 | $x^{4} - x + 27$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.3.2 | $x^{4} - 801$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |