Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 3 x + 4 x^{2} - 15 x^{3} + 25 x^{4}$ |
Frobenius angles: | $\pm0.0673911931187$, $\pm0.599275473548$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
Galois group: | $C_2^2$ |
Jacobians: | $3$ |
Isomorphism classes: | 3 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $12$ | $576$ | $11664$ | $361728$ | $9943932$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $25$ | $90$ | $577$ | $3183$ | $15478$ | $77619$ | $391777$ | $1954962$ | $9762625$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^5+3x^4+x^3+3x^2+2x$
- $y^2=3x^6+2x^5+2x^4+x^3+4x^2+x+2$
- $y^2=2x^5+x^4+x^2+x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{3}}$.
Endomorphism algebra over $\F_{5}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-11})\). |
The base change of $A$ to $\F_{5^{3}}$ is 1.125.as 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$ |
Base change
This is a primitive isogeny class.