Normalized defining polynomial
\( x^{20} - x^{19} - 8 x^{18} + 16 x^{17} + x^{16} - 86 x^{15} + 175 x^{14} + 169 x^{13} - 704 x^{12} + 71 x^{11} + 1178 x^{10} - 773 x^{9} - 770 x^{8} + 1039 x^{7} - 86 x^{6} - 417 x^{5} + 224 x^{4} - 7 x^{3} - 30 x^{2} + 10 x - 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(390615042633772247314453125=3^{6}\cdot 5^{15}\cdot 23^{4}\cdot 89^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $21.36$ | 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: | $3, 5, 23, 89$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{12012021693167} a^{19} - \frac{3313913308781}{12012021693167} a^{18} + \frac{4836429650102}{12012021693167} a^{17} - \frac{1479508150285}{12012021693167} a^{16} - \frac{5176327774430}{12012021693167} a^{15} + \frac{808901670777}{12012021693167} a^{14} - \frac{5960098196164}{12012021693167} a^{13} - \frac{3340407798535}{12012021693167} a^{12} - \frac{744651363615}{12012021693167} a^{11} - \frac{3006449752822}{12012021693167} a^{10} - \frac{5501777060118}{12012021693167} a^{9} - \frac{2378830949606}{12012021693167} a^{8} + \frac{3102407005896}{12012021693167} a^{7} + \frac{373164488670}{12012021693167} a^{6} + \frac{1694754697062}{12012021693167} a^{5} - \frac{4104414180843}{12012021693167} a^{4} - \frac{2349352565805}{12012021693167} a^{3} - \frac{5865876487873}{12012021693167} a^{2} - \frac{1822642472461}{12012021693167} a - \frac{5441977111308}{12012021693167}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 1185515.34399 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 7680 |
| The 72 conjugacy class representatives for t20n369 are not computed |
| Character table for t20n369 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1 |
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 | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.8.6.1 | $x^{8} + 9 x^{4} + 36$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
| 3.12.0.1 | $x^{12} - x^{4} - x^{3} - x^{2} + x - 1$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.12.11.2 | $x^{12} - 20$ | $12$ | $1$ | $11$ | $S_3 \times C_4$ | $[\ ]_{12}^{2}$ | |
| $23$ | 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.8.4.1 | $x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $89$ | 89.3.0.1 | $x^{3} - x + 7$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 89.3.0.1 | $x^{3} - x + 7$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 89.3.0.1 | $x^{3} - x + 7$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 89.3.0.1 | $x^{3} - x + 7$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 89.4.2.1 | $x^{4} + 979 x^{2} + 285156$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.4.2.1 | $x^{4} + 979 x^{2} + 285156$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |