Normalized defining polynomial
\( x^{21} - 29403 x^{14} + 214879311 x^{7} + 10460353203 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[3, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-996726852728145413540441359277571789889580380640583=-\,3^{18}\cdot 7^{29}\cdot 19^{14}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $268.23$ | 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: | $3, 7, 19$ | 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 \(a\))
$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{9} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{27} a^{5}$, $\frac{1}{81} a^{6}$, $\frac{1}{486} a^{7} - \frac{1}{2}$, $\frac{1}{486} a^{8} - \frac{1}{2} a$, $\frac{1}{1458} a^{9} - \frac{1}{6} a^{2}$, $\frac{1}{4374} a^{10} - \frac{1}{18} a^{3}$, $\frac{1}{91854} a^{11} + \frac{1}{30618} a^{10} + \frac{1}{3402} a^{9} - \frac{1}{1134} a^{8} - \frac{1}{3402} a^{7} + \frac{11}{378} a^{4} - \frac{1}{42} a^{3} + \frac{5}{42} a^{2} - \frac{5}{14} a + \frac{3}{14}$, $\frac{1}{275562} a^{12} + \frac{1}{15309} a^{10} + \frac{1}{10206} a^{9} - \frac{1}{1701} a^{8} + \frac{1}{3402} a^{7} + \frac{11}{1134} a^{5} - \frac{1}{21} a^{3} - \frac{1}{14} a^{2} + \frac{3}{7} a - \frac{3}{14}$, $\frac{1}{826686} a^{13} - \frac{1}{30618} a^{10} - \frac{1}{10206} a^{9} - \frac{1}{1134} a^{8} + \frac{1}{1701} a^{7} + \frac{11}{3402} a^{6} + \frac{1}{42} a^{3} + \frac{1}{14} a^{2} - \frac{5}{14} a - \frac{3}{7}$, $\frac{1}{4960116} a^{14} + \frac{2}{5103} a^{7} - \frac{11}{28}$, $\frac{1}{14880348} a^{15} + \frac{25}{30618} a^{8} + \frac{31}{84} a$, $\frac{1}{44641044} a^{16} + \frac{25}{91854} a^{9} + \frac{31}{252} a^{2}$, $\frac{1}{401769396} a^{17} + \frac{44}{413343} a^{10} - \frac{11}{2268} a^{3}$, $\frac{1}{1205308188} a^{18} + \frac{1}{354294} a^{11} - \frac{1}{10206} a^{10} - \frac{1}{5103} a^{9} + \frac{1}{1701} a^{8} + \frac{1}{1134} a^{7} - \frac{101}{6804} a^{4} - \frac{5}{126} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{5}{14}$, $\frac{1}{10847773692} a^{19} - \frac{10}{11160261} a^{12} + \frac{1}{15309} a^{10} + \frac{1}{10206} a^{9} - \frac{1}{1701} a^{8} + \frac{1}{3402} a^{7} + \frac{457}{61236} a^{5} - \frac{1}{21} a^{3} - \frac{1}{14} a^{2} + \frac{3}{7} a - \frac{3}{14}$, $\frac{1}{32543321076} a^{20} - \frac{10}{33480783} a^{13} - \frac{1}{30618} a^{10} - \frac{1}{10206} a^{9} - \frac{1}{1134} a^{8} + \frac{1}{1701} a^{7} + \frac{457}{183708} a^{6} + \frac{1}{42} a^{3} + \frac{1}{14} a^{2} - \frac{5}{14} a - \frac{3}{7}$
Class group and class number
$C_{42}$, which has order $42$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 33794281381556120 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times F_7$ (as 21T9):
| A solvable group of order 126 |
| The 21 conjugacy class representatives for $C_3\times F_7$ |
| Character table for $C_3\times F_7$ is not computed |
Intermediate fields
| 3.3.17689.1, Deg 7 |
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 | ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ | R | $21$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ | R | ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.7.6.1 | $x^{7} - 3$ | $7$ | $1$ | $6$ | $F_7$ | $[\ ]_{7}^{6}$ |
| 3.7.6.1 | $x^{7} - 3$ | $7$ | $1$ | $6$ | $F_7$ | $[\ ]_{7}^{6}$ | |
| 3.7.6.1 | $x^{7} - 3$ | $7$ | $1$ | $6$ | $F_7$ | $[\ ]_{7}^{6}$ | |
| 7 | Data not computed | ||||||
| 19 | Data not computed | ||||||