Normalized defining polynomial
\( x^{20} - 2 x^{19} + 5 x^{18} + 60 x^{17} - 119 x^{16} - 84 x^{15} + 2577 x^{14} - 5921 x^{13} - 1119 x^{12} + 49587 x^{11} - 117590 x^{10} + 81686 x^{9} + 173576 x^{8} - 472480 x^{7} + 308850 x^{6} + 64335 x^{5} - 296334 x^{4} + 118979 x^{3} + 120463 x^{2} - 3930 x - 4889 \)
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: | \(1069756574259831337445098876953125=5^{15}\cdot 19^{5}\cdot 1699^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $44.82$ | 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: | $5, 19, 1699$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{29} a^{18} - \frac{14}{29} a^{17} - \frac{8}{29} a^{16} - \frac{7}{29} a^{15} - \frac{8}{29} a^{14} + \frac{3}{29} a^{13} - \frac{13}{29} a^{12} + \frac{14}{29} a^{11} - \frac{7}{29} a^{10} + \frac{12}{29} a^{9} - \frac{3}{29} a^{8} + \frac{3}{29} a^{7} - \frac{4}{29} a^{6} - \frac{14}{29} a^{5} - \frac{7}{29} a^{4} - \frac{8}{29} a^{3} - \frac{12}{29} a^{2} - \frac{11}{29} a + \frac{10}{29}$, $\frac{1}{1337985166006905283745608578145233322369978456984642840339} a^{19} + \frac{10352707755991034143451891531201154125893910463878619931}{1337985166006905283745608578145233322369978456984642840339} a^{18} + \frac{279753143663337127450219562986545123484698386239791325793}{1337985166006905283745608578145233322369978456984642840339} a^{17} - \frac{356865124452587444158120909070505121793795436280210039011}{1337985166006905283745608578145233322369978456984642840339} a^{16} - \frac{65343477798386311658694344362632326875610420863491056684}{1337985166006905283745608578145233322369978456984642840339} a^{15} + \frac{23820741992586303625557631778508100834798994404076287129}{1337985166006905283745608578145233322369978456984642840339} a^{14} - \frac{87006493794035378576298423733040083904159693932266508906}{1337985166006905283745608578145233322369978456984642840339} a^{13} - \frac{413113126062113000658832947332298828533700178062059125236}{1337985166006905283745608578145233322369978456984642840339} a^{12} + \frac{472495573597089771014112359880368412688946627771804942547}{1337985166006905283745608578145233322369978456984642840339} a^{11} + \frac{367552836666186351460104796760696992650689748024839290717}{1337985166006905283745608578145233322369978456984642840339} a^{10} - \frac{359270763640229101228823123750673008976088433318816250913}{1337985166006905283745608578145233322369978456984642840339} a^{9} - \frac{143206278380446606859642661280289154247126903924470847326}{1337985166006905283745608578145233322369978456984642840339} a^{8} - \frac{5289506136302571315001284803612297675157970666241656270}{1337985166006905283745608578145233322369978456984642840339} a^{7} + \frac{500278283594653257967064083394631241053483330499747324577}{1337985166006905283745608578145233322369978456984642840339} a^{6} + \frac{21816007463769733662240069191853329360806452419931442081}{1337985166006905283745608578145233322369978456984642840339} a^{5} - \frac{429338943512794433654944573936104536065214232939869635587}{1337985166006905283745608578145233322369978456984642840339} a^{4} + \frac{500502860721856724742822246199491611509202759595349923368}{1337985166006905283745608578145233322369978456984642840339} a^{3} + \frac{103543998146098645078442546358242691317158618902996412091}{1337985166006905283745608578145233322369978456984642840339} a^{2} + \frac{315275172734461851704183400255010465150211301921926974858}{1337985166006905283745608578145233322369978456984642840339} a + \frac{37646001978262214168094690216205139248111962707408140938}{1337985166006905283745608578145233322369978456984642840339}$
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: | \( 905470208.461 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 102400 |
| The 130 conjugacy class representatives for t20n771 are not computed |
| Character table for t20n771 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.10.3256446753125.1 |
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 | $20$ | $20$ | R | $20$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | R | $20$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | $20$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 | ||||||
| $19$ | $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.4.3.2 | $x^{4} - 19$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 1699 | Data not computed | ||||||