Normalized defining polynomial
\( x^{21} - 199917 x^{14} - 8633886580 x^{7} - 80798284478113 \)
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: | \(-1860948596841831834340148197569891276052713687190665695546023=-\,7^{29}\cdot 97^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $741.29$ | 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: | $7, 97$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{97} a^{7}$, $\frac{1}{97} a^{8}$, $\frac{1}{97} a^{9}$, $\frac{1}{97} a^{10}$, $\frac{1}{65863} a^{11} + \frac{3}{679} a^{10} + \frac{1}{679} a^{9} + \frac{3}{679} a^{8} - \frac{1}{679} a^{7} - \frac{218}{679} a^{4} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{65863} a^{12} + \frac{3}{679} a^{10} - \frac{1}{679} a^{9} + \frac{1}{679} a^{8} - \frac{3}{679} a^{7} - \frac{218}{679} a^{5} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{65863} a^{13} + \frac{1}{679} a^{10} - \frac{3}{679} a^{9} - \frac{1}{679} a^{8} - \frac{3}{679} a^{7} - \frac{218}{679} a^{6} - \frac{1}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{785932838509} a^{14} - \frac{40609074}{8102400397} a^{7} - \frac{313650}{861133}$, $\frac{1}{785932838509} a^{15} - \frac{40609074}{8102400397} a^{8} - \frac{313650}{861133} a$, $\frac{1}{76235485335373} a^{16} - \frac{1210027688}{785932838509} a^{9} + \frac{2269749}{83529901} a^{2}$, $\frac{1}{76235485335373} a^{17} - \frac{1210027688}{785932838509} a^{10} + \frac{2269749}{83529901} a^{3}$, $\frac{1}{7394842077531181} a^{18} - \frac{52541917}{76235485335373} a^{11} + \frac{3}{679} a^{10} + \frac{1}{679} a^{9} + \frac{3}{679} a^{8} - \frac{1}{679} a^{7} - \frac{478694162}{1157485771} a^{4} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7394842077531181} a^{19} - \frac{52541917}{76235485335373} a^{12} + \frac{3}{679} a^{10} - \frac{1}{679} a^{9} + \frac{1}{679} a^{8} - \frac{3}{679} a^{7} - \frac{478694162}{1157485771} a^{5} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{717299681520524557} a^{20} - \frac{50981915841}{7394842077531181} a^{13} + \frac{1}{679} a^{10} - \frac{3}{679} a^{9} - \frac{1}{679} a^{8} - \frac{3}{679} a^{7} - \frac{99553439400}{785932838509} a^{6} - \frac{1}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{3}{7}$
Class group and class number
Not computed
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: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times D_7$ (as 21T3):
| A solvable group of order 42 |
| The 15 conjugacy class representatives for $C_3\times D_7$ |
| Character table for $C_3\times D_7$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), Deg 7 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | 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 | $21$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ | R | $21$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ | $21$ | ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | $21$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
| 7 | Data not computed | ||||||
| $97$ | 97.7.6.1 | $x^{7} - 97$ | $7$ | $1$ | $6$ | $D_{7}$ | $[\ ]_{7}^{2}$ |
| 97.7.6.1 | $x^{7} - 97$ | $7$ | $1$ | $6$ | $D_{7}$ | $[\ ]_{7}^{2}$ | |
| 97.7.6.1 | $x^{7} - 97$ | $7$ | $1$ | $6$ | $D_{7}$ | $[\ ]_{7}^{2}$ | |