Invariants
Base field: | $\F_{7}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 5 x + 7 x^{2} )( 1 - 5 x + 17 x^{2} - 35 x^{3} + 49 x^{4} )$ |
$1 - 10 x + 49 x^{2} - 155 x^{3} + 343 x^{4} - 490 x^{5} + 343 x^{6}$ | |
Frobenius angles: | $\pm0.106147807505$, $\pm0.197751856397$, $\pm0.457936209148$ |
Angle rank: | $3$ (numerical) |
Isomorphism classes: | 11 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $81$ | $112671$ | $41176836$ | $13670259759$ | $4783070555856$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $48$ | $349$ | $2372$ | $16933$ | $118989$ | $826894$ | $5765492$ | $40344613$ | $282483003$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7}$.
Endomorphism algebra over $\F_{7}$The isogeny class factors as 1.7.af $\times$ 2.7.af_r and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.