Normalized defining polynomial
\( x^{21} - 9 x^{14} - 12 x^{7} + 1 \)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[3, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(-108608979330127274981177223\)\(\medspace = -\,3^{28}\cdot 7^{15}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $17.37$ | 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: | $3, 7$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $3$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} + \frac{3}{17} a^{7} + \frac{7}{17}$, $\frac{1}{17} a^{15} + \frac{3}{17} a^{8} + \frac{7}{17} a$, $\frac{1}{17} a^{16} + \frac{3}{17} a^{9} + \frac{7}{17} a^{2}$, $\frac{1}{17} a^{17} + \frac{3}{17} a^{10} + \frac{7}{17} a^{3}$, $\frac{1}{119} a^{18} + \frac{2}{119} a^{17} + \frac{2}{119} a^{16} - \frac{1}{119} a^{15} - \frac{1}{119} a^{14} - \frac{3}{7} a^{13} + \frac{2}{7} a^{12} - \frac{48}{119} a^{11} + \frac{6}{119} a^{10} - \frac{45}{119} a^{9} - \frac{54}{119} a^{8} - \frac{3}{119} a^{7} - \frac{1}{7} a^{5} + \frac{41}{119} a^{4} - \frac{3}{119} a^{3} + \frac{48}{119} a^{2} - \frac{58}{119} a - \frac{41}{119}$, $\frac{1}{119} a^{19} - \frac{2}{119} a^{17} + \frac{2}{119} a^{16} + \frac{1}{119} a^{15} + \frac{1}{7} a^{13} + \frac{3}{119} a^{12} - \frac{1}{7} a^{11} - \frac{57}{119} a^{10} + \frac{57}{119} a^{9} - \frac{2}{17} a^{8} + \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{44}{119} a^{5} + \frac{2}{7} a^{4} + \frac{54}{119} a^{3} + \frac{2}{17} a^{2} - \frac{44}{119} a - \frac{3}{7}$, $\frac{1}{119} a^{20} - \frac{1}{119} a^{17} - \frac{2}{119} a^{16} - \frac{2}{119} a^{15} + \frac{1}{119} a^{14} + \frac{20}{119} a^{13} + \frac{3}{7} a^{12} - \frac{2}{7} a^{11} + \frac{48}{119} a^{10} - \frac{6}{119} a^{9} + \frac{45}{119} a^{8} + \frac{54}{119} a^{7} - \frac{44}{119} a^{6} + \frac{1}{7} a^{4} - \frac{41}{119} a^{3} + \frac{3}{119} a^{2} - \frac{48}{119} a + \frac{58}{119}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | 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 (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 51151.7616302 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
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
| \(\Q(\zeta_{9})^+\), 7.1.110270727.1 |
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 | $21$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 7 | Data not computed | ||||||