Normalized defining polynomial
\( x^{20} - 21 x^{18} + 221 x^{16} - 1520 x^{14} + 7113 x^{12} - 22369 x^{10} + 46399 x^{8} - 61610 x^{6} + 49225 x^{4} - 20625 x^{2} + 3125 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(19750040792380468310835200000000000=2^{36}\cdot 5^{11}\cdot 19^{4}\cdot 461^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $51.85$ | 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: | $2, 5, 19, 461$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{12} + \frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{13} + \frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{16} - \frac{1}{25} a^{14} + \frac{1}{25} a^{12} - \frac{12}{25} a^{8} - \frac{9}{25} a^{6} - \frac{6}{25} a^{4} - \frac{1}{5} a^{2}$, $\frac{1}{125} a^{17} - \frac{1}{125} a^{15} - \frac{49}{125} a^{13} - \frac{12}{125} a^{9} + \frac{16}{125} a^{7} - \frac{31}{125} a^{5} + \frac{4}{25} a^{3}$, $\frac{1}{1738564125} a^{18} + \frac{27217094}{1738564125} a^{16} - \frac{34048073}{579521375} a^{14} - \frac{173364616}{347712825} a^{12} - \frac{485781337}{1738564125} a^{10} - \frac{62255833}{579521375} a^{8} + \frac{401356264}{1738564125} a^{6} - \frac{1134710}{4636171} a^{4} + \frac{9887734}{69542565} a^{2} - \frac{3208544}{13908513}$, $\frac{1}{8692820625} a^{19} + \frac{27217094}{8692820625} a^{17} + \frac{81856202}{2897606875} a^{15} - \frac{590620006}{1738564125} a^{13} - \frac{3615196762}{8692820625} a^{11} + \frac{517265542}{2897606875} a^{9} - \frac{3771197636}{8692820625} a^{7} - \frac{47399089}{115904275} a^{5} - \frac{143105909}{347712825} a^{3} - \frac{6205114}{13908513} a$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 30290376786.9 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 3686400 |
| The 180 conjugacy class representatives for t20n1010 are not computed |
| Character table for t20n1010 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.6.61376064800000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 40 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ | R | $20$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ | $20$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 | Data not computed | ||||||
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.12.6.1 | $x^{12} + 500 x^{6} - 3125 x^{2} + 62500$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| $19$ | $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.5.0.1 | $x^{5} - x + 5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 19.5.0.1 | $x^{5} - x + 5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 461 | Data not computed | ||||||