Normalized defining polynomial
\( x^{20} - 3 x^{19} + 4 x^{18} - 5 x^{17} + 9 x^{16} - 8 x^{15} + 9 x^{14} - 16 x^{13} + 11 x^{12} - 10 x^{11} + 17 x^{10} - 10 x^{9} + 11 x^{8} - 16 x^{7} + 9 x^{6} - 8 x^{5} + 9 x^{4} - 5 x^{3} + 4 x^{2} - 3 x + 1 \)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(602207464968018231001\)\(\medspace = 47^{10}\cdot 107^{2}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $10.94$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $47, 107$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $4$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{29} a^{18} + \frac{10}{29} a^{17} - \frac{12}{29} a^{16} + \frac{3}{29} a^{15} + \frac{2}{29} a^{14} - \frac{14}{29} a^{13} - \frac{1}{29} a^{12} + \frac{14}{29} a^{11} - \frac{9}{29} a^{10} + \frac{4}{29} a^{9} - \frac{9}{29} a^{8} + \frac{14}{29} a^{7} - \frac{1}{29} a^{6} - \frac{14}{29} a^{5} + \frac{2}{29} a^{4} + \frac{3}{29} a^{3} - \frac{12}{29} a^{2} + \frac{10}{29} a + \frac{1}{29}$, $\frac{1}{377} a^{19} - \frac{4}{377} a^{18} + \frac{138}{377} a^{17} + \frac{2}{29} a^{16} - \frac{69}{377} a^{15} + \frac{74}{377} a^{14} - \frac{14}{29} a^{13} - \frac{146}{377} a^{12} + \frac{27}{377} a^{11} - \frac{102}{377} a^{10} + \frac{80}{377} a^{9} + \frac{53}{377} a^{8} - \frac{81}{377} a^{7} - \frac{121}{377} a^{5} + \frac{178}{377} a^{4} + \frac{7}{29} a^{3} - \frac{83}{377} a^{2} + \frac{35}{377} a + \frac{131}{377}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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 | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 79.7167288943 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_2\times C_2^4:D_5$ (as 20T81):
| 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
| \(\Q(\sqrt{-47}) \), 5.1.2209.1 x5, 10.2.24539915749.1, 10.0.522125867.1, 10.0.229345007.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ | R | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $47$ | 47.2.1.2 | $x^{2} + 94$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 47.2.1.2 | $x^{2} + 94$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 94$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 94$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 94$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 94$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.4.2.1 | $x^{4} + 1175 x^{2} + 373321$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.2.1 | $x^{4} + 1175 x^{2} + 373321$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $107$ | 107.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 107.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 107.4.2.1 | $x^{4} + 963 x^{2} + 286225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 107.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |