Normalized defining polynomial
\( x^{20} - 60 x^{18} - 92 x^{17} + 1149 x^{16} + 2729 x^{15} - 9680 x^{14} - 31471 x^{13} + 37687 x^{12} + 186305 x^{11} - 47867 x^{10} - 623685 x^{9} - 110356 x^{8} + 1208701 x^{7} + 469677 x^{6} - 1316497 x^{5} - 663935 x^{4} + 733746 x^{3} + 423836 x^{2} - 158296 x - 99181 \)
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: | \(-10685760975970208109485168457031250000=-\,2^{4}\cdot 5^{19}\cdot 11^{6}\cdot 71^{4}\cdot 167^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $71.03$ | 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: | $2, 5, 11, 71, 167$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2785570790802010425608426140141153439722} a^{19} - \frac{30566813379686080681738695521803902519}{2785570790802010425608426140141153439722} a^{18} + \frac{303668242504180300540820786312022343149}{1392785395401005212804213070070576719861} a^{17} + \frac{174118970004757554915729898525468967593}{2785570790802010425608426140141153439722} a^{16} - \frac{71231511704721029315413722015924965829}{1392785395401005212804213070070576719861} a^{15} - \frac{52386584747047237698855023172990913871}{107137338107769631754170236159275132297} a^{14} - \frac{26519568886697622225680904301870308303}{1392785395401005212804213070070576719861} a^{13} - \frac{1357751151798500068783621504593720791169}{2785570790802010425608426140141153439722} a^{12} - \frac{168928255626180780945566665615541001169}{1392785395401005212804213070070576719861} a^{11} - \frac{554841423732556286604093223039744192672}{1392785395401005212804213070070576719861} a^{10} - \frac{96007634177692879605410196012667289539}{214274676215539263508340472318550264594} a^{9} + \frac{179232922483453645954713635628258701113}{1392785395401005212804213070070576719861} a^{8} + \frac{795580412226434123458158707893039878323}{2785570790802010425608426140141153439722} a^{7} - \frac{368995283893754306131661066736089113205}{1392785395401005212804213070070576719861} a^{6} - \frac{299659262259865369854142057514254199969}{1392785395401005212804213070070576719861} a^{5} - \frac{575082088418344858112644696473077279665}{2785570790802010425608426140141153439722} a^{4} + \frac{264865200829432536084117478252138436808}{1392785395401005212804213070070576719861} a^{3} + \frac{578865770054491807711362429387386806203}{2785570790802010425608426140141153439722} a^{2} + \frac{936818800536976916467746399040179967145}{2785570790802010425608426140141153439722} a - \frac{189382080278353861851688039952215210541}{1392785395401005212804213070070576719861}$
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: | \( 1613412498980 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 1857945600 |
| The 260 conjugacy class representatives for t20n1106 are not computed |
| Character table for t20n1106 is not computed |
Intermediate fields
| 10.10.6645000909765625.1 |
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 | ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.8.0.1}{8} }$ | R | ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ | R | $18{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | $18{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.4.4 | $x^{4} - 5$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
| 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $5$ | 5.4.2.2 | $x^{4} - 5 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 5.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 5.10.17.7 | $x^{10} - 15 x^{8} + 5$ | $10$ | $1$ | $17$ | $F_{5}\times C_2$ | $[2]_{2}^{4}$ | |
| $11$ | 11.6.0.1 | $x^{6} + x^{2} - 2 x + 8$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 11.6.0.1 | $x^{6} + x^{2} - 2 x + 8$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 11.8.6.3 | $x^{8} - 11 x^{4} + 847$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ | |
| $71$ | 71.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 71.4.2.1 | $x^{4} + 1491 x^{2} + 609961$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 71.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 71.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 71.4.2.1 | $x^{4} + 1491 x^{2} + 609961$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $167$ | $\Q_{167}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{167}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 167.4.0.1 | $x^{4} - x + 60$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 167.6.0.1 | $x^{6} - x + 23$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 167.8.4.1 | $x^{8} + 3346680 x^{4} - 4657463 x^{2} + 2800066755600$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |