Normalized defining polynomial
\( x^{20} - 4 x^{19} + 12 x^{18} - 104 x^{17} + 258 x^{16} + 378 x^{15} - 948 x^{14} - 5664 x^{13} + 15540 x^{12} + 14907 x^{11} - 134419 x^{10} + 301534 x^{9} - 297764 x^{8} - 169134 x^{7} + 941416 x^{6} - 1311498 x^{5} + 1032859 x^{4} - 582288 x^{3} + 241614 x^{2} - 27090 x - 22131 \)
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: | \(194813520645626483248608015673828125=3^{8}\cdot 5^{11}\cdot 239^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $58.14$ | 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: | $3, 5, 239$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{15} a^{18} - \frac{7}{15} a^{17} - \frac{1}{5} a^{16} + \frac{7}{15} a^{15} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{3} a^{8} + \frac{1}{15} a^{7} - \frac{2}{15} a^{6} + \frac{2}{5} a^{5} - \frac{1}{3} a^{4} + \frac{2}{5} a^{3} + \frac{1}{15} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{1958353661342093027374925436777814284138575706886930965} a^{19} - \frac{57889708846272463758734681350313341048574427296840657}{1958353661342093027374925436777814284138575706886930965} a^{18} - \frac{731023274504910096823823659534971059796313532045211923}{1958353661342093027374925436777814284138575706886930965} a^{17} - \frac{591346466025388754067125001886011086923002109590238043}{1958353661342093027374925436777814284138575706886930965} a^{16} - \frac{62871229080709497438682701001146970423184502659332880}{391670732268418605474985087355562856827715141377386193} a^{15} + \frac{222837834713513976202560079817158104703790273805056392}{652784553780697675791641812259271428046191902295643655} a^{14} - \frac{156290712414575087595027581249582948203087540334011222}{652784553780697675791641812259271428046191902295643655} a^{13} + \frac{69476975331553257160504253186928202712820006910471451}{652784553780697675791641812259271428046191902295643655} a^{12} - \frac{35013968919795959339913937896027027778669254898176041}{652784553780697675791641812259271428046191902295643655} a^{11} + \frac{169074620305449520392287646347657420713893879918691406}{652784553780697675791641812259271428046191902295643655} a^{10} + \frac{114173008971910523540182303551303254009745033509955616}{391670732268418605474985087355562856827715141377386193} a^{9} + \frac{82294396083990383620322402314401000750029555403715646}{1958353661342093027374925436777814284138575706886930965} a^{8} + \frac{125953314275017380249603779292686970946319227518086161}{652784553780697675791641812259271428046191902295643655} a^{7} - \frac{90020487556946189518010995546439359151851932933406129}{1958353661342093027374925436777814284138575706886930965} a^{6} - \frac{173461650954092658481875325340496056393448220712426971}{391670732268418605474985087355562856827715141377386193} a^{5} + \frac{157350117242565614097699986315262685265642499227020106}{1958353661342093027374925436777814284138575706886930965} a^{4} + \frac{49960770934530097330623637651134265247188129788484876}{1958353661342093027374925436777814284138575706886930965} a^{3} - \frac{437090964126698866746048532053434469743783288227611937}{1958353661342093027374925436777814284138575706886930965} a^{2} + \frac{183538020874732806665434240573927185792836316714276723}{652784553780697675791641812259271428046191902295643655} a - \frac{25249856176336311960816546948326197575364890711926183}{130556910756139535158328362451854285609238380459128731}$
Class group and class number
$C_{2}$, which has order $2$ (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: | \( 3139974169.35 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 10240 |
| The 100 conjugacy class representatives for t20n426 are not computed |
| Character table for t20n426 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1 |
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 | $20$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ | $20$ | $20$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 239 | Data not computed | ||||||