Normalized defining polynomial
\( x^{20} - x^{19} - 48 x^{18} + 43 x^{17} + 915 x^{16} - 876 x^{15} - 9157 x^{14} + 10182 x^{13} + 52575 x^{12} - 70067 x^{11} - 174991 x^{10} + 283398 x^{9} + 317729 x^{8} - 639806 x^{7} - 254704 x^{6} + 701537 x^{5} + 19456 x^{4} - 249175 x^{3} + 23965 x^{2} + 12130 x - 295 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[18, 1]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-154944594075003493045619628906250000=-\,2^{4}\cdot 5^{15}\cdot 19^{4}\cdot 29^{7}\cdot 109^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $57.48$ | 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: | $2, 5, 19, 29, 109$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{98890861901110095021481121446973485689729} a^{19} - \frac{12143957828644390518819064893167198852744}{98890861901110095021481121446973485689729} a^{18} - \frac{3730685480528367192383247214620831126439}{98890861901110095021481121446973485689729} a^{17} + \frac{21503548655568922257212809486169837258867}{98890861901110095021481121446973485689729} a^{16} + \frac{31006564486736838268324585385599083969451}{98890861901110095021481121446973485689729} a^{15} - \frac{11063735164252083093508199758118539130428}{98890861901110095021481121446973485689729} a^{14} + \frac{363720539244995760902349164673244140840}{32963620633703365007160373815657828563243} a^{13} - \frac{2835411093299976773799804611213900102631}{32963620633703365007160373815657828563243} a^{12} + \frac{14523034995492511370159659779416372702516}{32963620633703365007160373815657828563243} a^{11} + \frac{34843222264393650255280353507020447908261}{98890861901110095021481121446973485689729} a^{10} - \frac{39004886416628794979819929395771202056191}{98890861901110095021481121446973485689729} a^{9} + \frac{23940418182897603457147183615342976003999}{98890861901110095021481121446973485689729} a^{8} - \frac{10291294087865174892211334900709063861747}{32963620633703365007160373815657828563243} a^{7} - \frac{25428593468631001589389489968276048123725}{98890861901110095021481121446973485689729} a^{6} + \frac{22371084403324683757310544424997999592256}{98890861901110095021481121446973485689729} a^{5} + \frac{47260913167882933813216920679365458568160}{98890861901110095021481121446973485689729} a^{4} + \frac{12559340241129207236268808920991801387919}{32963620633703365007160373815657828563243} a^{3} + \frac{14704216329290643061328551144991447645426}{98890861901110095021481121446973485689729} a^{2} + \frac{18950713670233054326771408770740395092501}{98890861901110095021481121446973485689729} a + \frac{38764597123519931105055671015902344795188}{98890861901110095021481121446973485689729}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $18$ | 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: | \( 162589955704 \) (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
| \(\Q(\sqrt{5}) \), 10.10.8172298511640625.1 |
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 | $16{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | $20$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ | $16{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.4.3 | $x^{4} + 2 x^{2} + 4 x + 4$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
| 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 5 | Data not computed | ||||||
| $19$ | 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.4.2.2 | $x^{4} - 19 x^{2} + 722$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.2 | $x^{4} - 19 x^{2} + 722$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 19.10.0.1 | $x^{10} + x^{2} - 2 x + 14$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| $29$ | $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.8.6.2 | $x^{8} + 145 x^{4} + 7569$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $109$ | $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 109.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.4.2.1 | $x^{4} + 1199 x^{2} + 427716$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 109.4.2.1 | $x^{4} + 1199 x^{2} + 427716$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |