Normalized defining polynomial
\( x^{20} - 10 x^{17} + 10 x^{16} + 14 x^{15} + 55 x^{14} + 160 x^{13} - 150 x^{12} + 30 x^{11} - 1821 x^{10} + 2060 x^{9} - 1635 x^{8} + 820 x^{7} + 2350 x^{6} - 2726 x^{5} + 245 x^{4} + 540 x^{3} - 80 x^{2} - 40 x - 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1349292851200000000000000000000=2^{34}\cdot 5^{20}\cdot 7^{7}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $32.10$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{280686495304184634885820422447271} a^{19} - \frac{38454362604542062136331302945653}{280686495304184634885820422447271} a^{18} + \frac{15498436199378101113153299816894}{280686495304184634885820422447271} a^{17} + \frac{64172180060548548042669199709629}{280686495304184634885820422447271} a^{16} + \frac{113557670470421013347880764101600}{280686495304184634885820422447271} a^{15} - \frac{131554393950350048886263720669778}{280686495304184634885820422447271} a^{14} - \frac{98872989908365637755621874026195}{280686495304184634885820422447271} a^{13} + \frac{54219919300350609874924596357438}{280686495304184634885820422447271} a^{12} - \frac{29502665098769560997646798813898}{280686495304184634885820422447271} a^{11} - \frac{113547129695960496743741262449693}{280686495304184634885820422447271} a^{10} + \frac{49150337189755569626558885623104}{280686495304184634885820422447271} a^{9} + \frac{69677860143712903569249125398998}{280686495304184634885820422447271} a^{8} + \frac{15471255621696529847756703080841}{280686495304184634885820422447271} a^{7} + \frac{128515958039003834442858819633917}{280686495304184634885820422447271} a^{6} - \frac{97905307559551585947636048325724}{280686495304184634885820422447271} a^{5} + \frac{102152920716493922963378046870101}{280686495304184634885820422447271} a^{4} + \frac{127434964724961766834157963930876}{280686495304184634885820422447271} a^{3} - \frac{48039435011052811036795813456462}{280686495304184634885820422447271} a^{2} - \frac{83917524252398513746226858498560}{280686495304184634885820422447271} a - \frac{63931015635442004151163766993604}{280686495304184634885820422447271}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 66074571.3319 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 409600 |
| The 190 conjugacy class representatives for t20n925 are not computed |
| Character table for t20n925 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 10.6.15680000000000.1 |
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 | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | R | $20$ | $20$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $7$ | 7.2.1.1 | $x^{2} - 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.8.6.3 | $x^{8} - 7 x^{4} + 147$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ | |