Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 3 x + 2 x^{2} - 21 x^{3} + 49 x^{4}$ |
Frobenius angles: | $\pm0.0252087988121$, $\pm0.641457867855$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-3}, \sqrt{-19})\) |
Galois group: | $C_2^2$ |
Jacobians: | $1$ |
Isomorphism classes: | 2 |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $28$ | $2128$ | $94864$ | $5592384$ | $281904868$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $45$ | $272$ | $2329$ | $16775$ | $116430$ | $821945$ | $5765329$ | $40334384$ | $282442725$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+3x^5+5x^4+6x^3+3x^2+3x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{3}}$.
Endomorphism algebra over $\F_{7}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-19})\). |
The base change of $A$ to $\F_{7^{3}}$ is 1.343.abk 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$ |
Base change
This is a primitive isogeny class.