Normalized defining polynomial
\( x^{20} - 10 x^{19} + 51 x^{18} - 174 x^{17} + 436 x^{16} - 836 x^{15} + 1246 x^{14} - 1436 x^{13} + 1232 x^{12} - 684 x^{11} + 56 x^{10} + 374 x^{9} - 472 x^{8} + 290 x^{7} - 18 x^{6} - 151 x^{5} + 123 x^{4} - 15 x^{3} - 34 x^{2} + 21 x + 9 \)
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: | \(11000468465851520789524462890625=5^{11}\cdot 19^{8}\cdot 29^{7}\cdot 769\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $35.65$ | 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: | $5, 19, 29, 769$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{939} a^{18} - \frac{3}{313} a^{17} - \frac{46}{313} a^{16} + \frac{123}{313} a^{15} + \frac{292}{939} a^{14} - \frac{295}{939} a^{13} + \frac{12}{313} a^{12} + \frac{55}{939} a^{11} + \frac{147}{313} a^{10} + \frac{62}{313} a^{9} - \frac{262}{939} a^{8} + \frac{436}{939} a^{7} + \frac{58}{313} a^{6} - \frac{79}{939} a^{5} - \frac{430}{939} a^{4} - \frac{446}{939} a^{3} + \frac{79}{939} a^{2} - \frac{410}{939} a + \frac{120}{313}$, $\frac{1}{10329} a^{19} - \frac{4}{10329} a^{18} + \frac{878}{3443} a^{17} + \frac{832}{3443} a^{16} - \frac{3497}{10329} a^{15} + \frac{226}{10329} a^{14} - \frac{500}{10329} a^{13} + \frac{1174}{10329} a^{12} - \frac{1162}{10329} a^{11} + \frac{44}{313} a^{10} + \frac{1607}{10329} a^{9} - \frac{874}{10329} a^{8} + \frac{3293}{10329} a^{7} - \frac{148}{10329} a^{6} - \frac{588}{3443} a^{5} + \frac{221}{10329} a^{4} - \frac{1343}{3443} a^{3} + \frac{28}{313} a^{2} + \frac{4883}{10329} a + \frac{1226}{3443}$
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: | \( 44070602.0243 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 819200 |
| The 275 conjugacy class representatives for t20n955 are not computed |
| Character table for t20n955 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.6.9932496465625.1 |
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 | $16{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ | $16{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | R | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| $19$ | 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.8.0.1 | $x^{8} - x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 19.10.8.1 | $x^{10} - 209 x^{5} + 11552$ | $5$ | $2$ | $8$ | $D_5$ | $[\ ]_{5}^{2}$ | |
| $29$ | $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.4.3.4 | $x^{4} + 232$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 29.4.3.4 | $x^{4} + 232$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 29.8.0.1 | $x^{8} + x^{2} - 3 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 769 | Data not computed | ||||||