Normalized defining polynomial
\( x^{20} - 14 x^{18} + 62 x^{16} - 16 x^{15} - 104 x^{14} + 196 x^{13} + 177 x^{12} - 696 x^{11} - 770 x^{10} + 728 x^{9} + 1382 x^{8} + 56 x^{7} - 900 x^{6} - 348 x^{5} + 242 x^{4} + 168 x^{3} - 16 x - 2 \)
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: | \(4991699275727550043990996300070912=2^{54}\cdot 31^{4}\cdot 113\cdot 227^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $48.41$ | 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: | $2, 31, 113, 227$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{222247132458401} a^{19} - \frac{26422248955132}{222247132458401} a^{18} + \frac{69653121808580}{222247132458401} a^{17} + \frac{102796709090871}{222247132458401} a^{16} + \frac{4586835714476}{222247132458401} a^{15} - \frac{5638591864183}{222247132458401} a^{14} + \frac{31934120561345}{222247132458401} a^{13} + \frac{58568767823845}{222247132458401} a^{12} + \frac{25749105604376}{222247132458401} a^{11} + \frac{5443795435927}{222247132458401} a^{10} + \frac{109725680666422}{222247132458401} a^{9} - \frac{39405404948528}{222247132458401} a^{8} - \frac{103025084046453}{222247132458401} a^{7} - \frac{53752936696932}{222247132458401} a^{6} + \frac{10850347805974}{222247132458401} a^{5} - \frac{71461039940890}{222247132458401} a^{4} - \frac{57618080524672}{222247132458401} a^{3} + \frac{45793635416407}{222247132458401} a^{2} - \frac{79340684881967}{222247132458401} a + \frac{534408625225}{222247132458401}$
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: | \( 12871883277.5 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 14745600 |
| The 384 conjugacy class representatives for t20n1037 are not computed |
| Character table for t20n1037 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 10.10.207699287474176.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ | ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ | R | $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | $16{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.26.1 | $x^{8} + 4 x^{6} + 8 x^{3} + 8 x^{2} + 2$ | $8$ | $1$ | $26$ | $C_2^2:C_4$ | $[2, 3, 7/2, 4]$ |
| 2.12.28.255 | $x^{12} + 2 x^{10} + 2 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{2} - 2$ | $12$ | $1$ | $28$ | $C_2 \times S_4$ | $[8/3, 8/3, 3]_{3}^{2}$ | |
| 31 | Data not computed | ||||||
| $113$ | 113.2.0.1 | $x^{2} - x + 10$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 113.2.0.1 | $x^{2} - x + 10$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.1.2 | $x^{2} + 339$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 113.4.0.1 | $x^{4} - x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 113.5.0.1 | $x^{5} - x + 17$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 113.5.0.1 | $x^{5} - x + 17$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 227 | Data not computed | ||||||