Normalized defining polynomial
\( x^{20} - 5080 x^{12} - 17552 x^{10} + 840350 x^{4} + 4033680 x^{2} + 3764768 \)
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: | \(4168598900368303957126361907200000000000000000000=2^{87}\cdot 5^{20}\cdot 7^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $269.77$ | 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: | $2, 5, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{2}{7} a^{4} - \frac{3}{7} a^{2}$, $\frac{1}{7} a^{13} + \frac{2}{7} a^{5} - \frac{3}{7} a^{3}$, $\frac{1}{49} a^{14} + \frac{16}{49} a^{6} - \frac{10}{49} a^{4}$, $\frac{1}{343} a^{15} + \frac{65}{343} a^{7} - \frac{59}{343} a^{5}$, $\frac{1}{686} a^{16} - \frac{139}{343} a^{8} + \frac{142}{343} a^{6}$, $\frac{1}{9604} a^{17} + \frac{1131}{2401} a^{9} + \frac{414}{2401} a^{7} - \frac{1}{2} a$, $\frac{1}{6787574627585590985832} a^{18} + \frac{57971135854733965}{121206689778314124747} a^{16} + \frac{1085679758730719}{2473605913843145403} a^{14} + \frac{158944991743116424}{2473605913843145403} a^{12} + \frac{11978695951323069137}{94271869827577652581} a^{10} - \frac{382288673796859639582}{848446828448198873229} a^{8} - \frac{39258431234824594361}{121206689778314124747} a^{6} + \frac{197355295074939026}{2473605913843145403} a^{4} + \frac{1947671033649500297}{9894423655372581612} a^{2} - \frac{44609454210559928}{353372273406163629}$, $\frac{1}{95026044786198273801648} a^{19} + \frac{57971135854733965}{1696893656896397746458} a^{17} - \frac{172886257547524298}{121206689778314124747} a^{15} + \frac{112316720383412210}{2473605913843145403} a^{13} + \frac{200522435606478374299}{1319806177586087136134} a^{11} - \frac{1039591165346628693020}{5939127799137392112603} a^{9} + \frac{384805351295941976210}{848446828448198873229} a^{7} - \frac{56908937032442384936}{121206689778314124747} a^{5} - \frac{4971863628655931017}{19788847310745163224} a^{3} + \frac{47295363757269095}{353372273406163629} a$
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: | \( 144511793412000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 14745600 |
| The 378 conjugacy class representatives for t20n1039 are not computed |
| Character table for t20n1039 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 10.6.1409873346560000000000.1 |
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 | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | R | R | $16{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | $20$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ | $20$ | $20$ |
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 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.2.1.1 | $x^{2} - 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.1 | $x^{2} - 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.10.8.1 | $x^{10} - 7 x^{5} + 147$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ | |