Normalized defining polynomial
\( x^{20} - 5 x^{19} - 9 x^{18} + 99 x^{17} - 266 x^{16} + 246 x^{15} + 1037 x^{14} - 6171 x^{13} + 12565 x^{12} - 4754 x^{11} - 36867 x^{10} + 81703 x^{9} - 75794 x^{8} - 57441 x^{7} - 94946 x^{6} + 601986 x^{5} - 1223688 x^{4} - 256048 x^{3} - 143581 x^{2} + 3058945 x - 999623 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[6, 7]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-785748055861331850537130300409483=-\,13^{10}\cdot 97^{2}\cdot 347^{7}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $44.13$ | 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: | $13, 97, 347$ | 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}{22809643086500207576107213276044543224865464587135478768893679616287} a^{19} - \frac{5191627668366679136722705296833752829638266274372935729435079031033}{22809643086500207576107213276044543224865464587135478768893679616287} a^{18} - \frac{3473452352815440833742188824424353039419634511433663232982888605670}{22809643086500207576107213276044543224865464587135478768893679616287} a^{17} - \frac{3627242931353231653582832704871902216031231269223258117195917117878}{22809643086500207576107213276044543224865464587135478768893679616287} a^{16} - \frac{1310451329809472119952759921825269106631324514225934460686161336679}{22809643086500207576107213276044543224865464587135478768893679616287} a^{15} - \frac{5337885733474579738059746698789618915425325705807299065243113169565}{22809643086500207576107213276044543224865464587135478768893679616287} a^{14} + \frac{7699731341779345243177291386779689131965019673852268606909338840016}{22809643086500207576107213276044543224865464587135478768893679616287} a^{13} + \frac{2413147752958247615071128432804490156951164655307577468710197826863}{22809643086500207576107213276044543224865464587135478768893679616287} a^{12} + \frac{10698287644112234530680012888486996566820075024634642921019411362537}{22809643086500207576107213276044543224865464587135478768893679616287} a^{11} + \frac{1465686361442451038286967826961736553722179298381617396503368283911}{22809643086500207576107213276044543224865464587135478768893679616287} a^{10} - \frac{8973659173724357020856106925358267855256888537008721874544848282686}{22809643086500207576107213276044543224865464587135478768893679616287} a^{9} + \frac{7944173158733133506712033147453728505435758266577989705578105926104}{22809643086500207576107213276044543224865464587135478768893679616287} a^{8} - \frac{6824738577936475327184186239005346864113395677426526437853417948658}{22809643086500207576107213276044543224865464587135478768893679616287} a^{7} + \frac{8806094936197182625060555050129593482799048973753379995183403047951}{22809643086500207576107213276044543224865464587135478768893679616287} a^{6} + \frac{7997683594146757774336556696341466390872896897302118778743871132743}{22809643086500207576107213276044543224865464587135478768893679616287} a^{5} + \frac{2176418270082783837990126145477512953074280924870640872600864740682}{22809643086500207576107213276044543224865464587135478768893679616287} a^{4} - \frac{10927649505568283598017701441526624834666117612119128570800036414935}{22809643086500207576107213276044543224865464587135478768893679616287} a^{3} - \frac{106071868823086190801686533794597174653950162580954079982930013214}{1341743710970600445653365486826149601462674387478557574640804683311} a^{2} + \frac{7172297588600479588264453272619218654095194705481035330153522924626}{22809643086500207576107213276044543224865464587135478768893679616287} a - \frac{6804792868972226541148247185366900726693257756562823097550985055983}{22809643086500207576107213276044543224865464587135478768893679616287}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $12$ | 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: | \( 573716183.695 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 122880 |
| The 108 conjugacy class representatives for t20n796 are not computed |
| Character table for t20n796 is not computed |
Intermediate fields
| \(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1 |
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 | $20$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $13$ | 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $97$ | 97.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 97.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.4.0.1 | $x^{4} - x + 23$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 97.4.2.1 | $x^{4} + 873 x^{2} + 235225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 97.4.0.1 | $x^{4} - x + 23$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 97.4.0.1 | $x^{4} - x + 23$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 347 | Data not computed | ||||||