Normalized defining polynomial
\( x^{20} - 4 x^{19} + 9 x^{18} - 23 x^{17} - 73 x^{16} + 128 x^{15} - 67 x^{14} + 318 x^{13} + 1428 x^{12} + 2366 x^{11} + 4792 x^{10} + 8162 x^{9} + 13098 x^{8} + 17165 x^{7} + 17760 x^{6} + 25222 x^{5} + 32482 x^{4} + 30702 x^{3} + 20825 x^{2} + 9261 x + 2401 \)
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: | \(5834302965409512291058019417402368=2^{10}\cdot 19^{10}\cdot 43^{11}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $48.79$ | 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: | $2, 19, 43$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7} a^{17} + \frac{3}{7} a^{16} + \frac{2}{7} a^{15} - \frac{2}{7} a^{14} - \frac{3}{7} a^{13} + \frac{2}{7} a^{12} + \frac{3}{7} a^{11} + \frac{3}{7} a^{10} - \frac{3}{7} a^{7} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{4067} a^{18} - \frac{11}{4067} a^{17} - \frac{943}{4067} a^{16} + \frac{1629}{4067} a^{15} - \frac{941}{4067} a^{14} + \frac{2011}{4067} a^{13} - \frac{1600}{4067} a^{12} + \frac{101}{4067} a^{11} - \frac{2}{581} a^{10} + \frac{30}{581} a^{9} + \frac{1460}{4067} a^{8} + \frac{30}{83} a^{7} + \frac{1191}{4067} a^{6} + \frac{988}{4067} a^{5} + \frac{1338}{4067} a^{4} - \frac{314}{4067} a^{3} - \frac{12}{4067} a^{2} + \frac{9}{581} a + \frac{4}{83}$, $\frac{1}{6003370508440492192183951849748732547977} a^{19} - \frac{251853146647156765846434582861077105}{6003370508440492192183951849748732547977} a^{18} - \frac{398164306683206245987220301332343145023}{6003370508440492192183951849748732547977} a^{17} + \frac{923552447962207213546417957914192036100}{6003370508440492192183951849748732547977} a^{16} + \frac{2538505486818693689559963244735770374471}{6003370508440492192183951849748732547977} a^{15} - \frac{44174459415253281076715726319950025101}{315966868865289062746523781565722765683} a^{14} - \frac{1649256185847109246013958072553180052064}{6003370508440492192183951849748732547977} a^{13} - \frac{7606947140995687792221406880852531758}{315966868865289062746523781565722765683} a^{12} - \frac{228523825964570554812797932759770451543}{857624358348641741740564549964104649711} a^{11} + \frac{111117081679999264762531166843746477}{1255672559807674585271690409903520717} a^{10} - \frac{61194303017720400266383499643634795470}{6003370508440492192183951849748732547977} a^{9} - \frac{3180119901922969898565891238344255898}{122517765478377391677223507137729235673} a^{8} + \frac{2574509265821046600828068731962832508951}{6003370508440492192183951849748732547977} a^{7} + \frac{844278678328910162640788050020401349118}{6003370508440492192183951849748732547977} a^{6} + \frac{2001974063188479614777785787577452271174}{6003370508440492192183951849748732547977} a^{5} - \frac{2942298724584100170827497917991739769748}{6003370508440492192183951849748732547977} a^{4} + \frac{956424993479612984383940533434110544381}{6003370508440492192183951849748732547977} a^{3} + \frac{278321198754421570633262332081123188088}{857624358348641741740564549964104649711} a^{2} - \frac{33916592543069241308566040957776456459}{122517765478377391677223507137729235673} a - \frac{323690047708555706487616017173450560}{17502537925482484525317643876818462239}$
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: | \( 857212252.196 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 10240 |
| The 160 conjugacy class representatives for t20n423 are not computed |
| Character table for t20n423 is not computed |
Intermediate fields
| \(\Q(\sqrt{817}) \), 5.5.667489.1 x5, 10.10.364007458703857.1 |
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 | ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | R | ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.10.10 | $x^{10} - 11 x^{8} + 10 x^{6} - 62 x^{4} + 21 x^{2} - 55$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ |
| 2.10.0.1 | $x^{10} - x^{3} + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| $19$ | 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $43$ | 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.4.3.1 | $x^{4} + 387$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 43.4.2.1 | $x^{4} + 215 x^{2} + 16641$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 43.4.2.1 | $x^{4} + 215 x^{2} + 16641$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 43.4.2.1 | $x^{4} + 215 x^{2} + 16641$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |