Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 7 x )^{2}( 1 - 8 x + 49 x^{2} )$ |
$1 - 22 x + 210 x^{2} - 1078 x^{3} + 2401 x^{4}$ | |
Frobenius angles: | $0$, $0$, $\pm0.306389419005$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1512$ | $5612544$ | $13838478696$ | $33226260480000$ | $79781820476078952$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $28$ | $2338$ | $117628$ | $5763646$ | $282438268$ | $13840846306$ | $678219946012$ | $33232917276286$ | $1628413575601372$ | $79792266286268578$ |
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 $\F_{7^{2}}$.
Endomorphism algebra over $\F_{7^{2}}$The isogeny class factors as 1.49.ao $\times$ 1.49.ai 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.