Normalized defining polynomial
\( x^{20} - 8 x^{19} + 17 x^{18} + 23 x^{17} - 138 x^{16} + 165 x^{15} - 49 x^{14} + 102 x^{13} - 128 x^{12} - 432 x^{11} + 855 x^{10} - 215 x^{9} - 589 x^{8} + 327 x^{7} + 338 x^{6} - 193 x^{5} - 144 x^{4} + 58 x^{3} - 8 x^{2} + 9 \)
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: | \(502465502329909572653052213761=37^{4}\cdot 401^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $30.55$ | 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: | $37, 401$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{93} a^{18} + \frac{26}{93} a^{17} - \frac{11}{93} a^{16} + \frac{8}{31} a^{15} + \frac{5}{31} a^{14} - \frac{3}{31} a^{13} + \frac{8}{93} a^{12} + \frac{26}{93} a^{11} - \frac{10}{31} a^{10} + \frac{13}{31} a^{9} - \frac{11}{31} a^{8} + \frac{16}{93} a^{7} + \frac{4}{31} a^{6} - \frac{4}{93} a^{4} + \frac{43}{93} a^{3} + \frac{37}{93} a^{2} + \frac{44}{93} a + \frac{5}{31}$, $\frac{1}{2304565836894744487689} a^{19} - \frac{3369228529022442936}{768188612298248162563} a^{18} - \frac{92381023688525533212}{768188612298248162563} a^{17} - \frac{448846646361219404075}{2304565836894744487689} a^{16} - \frac{290363557822853520093}{768188612298248162563} a^{15} - \frac{225283851528110319541}{768188612298248162563} a^{14} - \frac{149804831995221427645}{2304565836894744487689} a^{13} + \frac{785002550761317624307}{2304565836894744487689} a^{12} + \frac{452247517494414620909}{2304565836894744487689} a^{11} + \frac{11884564015331880428}{24780277816072521373} a^{10} - \frac{79378041049040086923}{768188612298248162563} a^{9} + \frac{44855411675854904350}{2304565836894744487689} a^{8} + \frac{1006343401341947442331}{2304565836894744487689} a^{7} + \frac{369527539547723615512}{768188612298248162563} a^{6} - \frac{823137178685520868225}{2304565836894744487689} a^{5} + \frac{349608231448755304556}{768188612298248162563} a^{4} - \frac{631464616190697906613}{2304565836894744487689} a^{3} + \frac{120239563119105944749}{768188612298248162563} a^{2} + \frac{272152014558449002922}{2304565836894744487689} a + \frac{188650833148954235656}{768188612298248162563}$
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: | \( 22330368.8867 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_2^4:D_5$ (as 20T87):
| A solvable group of order 320 |
| The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$ |
| Character table for $C_2\times C_2^4:D_5$ |
Intermediate fields
| 5.5.160801.1, 10.6.35398180431769.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| 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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $37$ | 37.4.2.1 | $x^{4} + 333 x^{2} + 34225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.2.1 | $x^{4} + 333 x^{2} + 34225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 401 | Data not computed | ||||||