Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 11 x^{2} - 20 x^{3} + 25 x^{4}$ |
Frobenius angles: | $\pm0.185749715683$, $\pm0.480916950984$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\zeta_{12})\) |
Galois group: | $C_2^2$ |
Jacobians: | $2$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $13$ | $793$ | $16900$ | $381433$ | $10005853$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $32$ | $134$ | $612$ | $3202$ | $16094$ | $78682$ | $389572$ | $1950254$ | $9765152$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+4x^5+3x^4+2x^3+4x^2+2x+2$
- $y^2=3x^6+4x^5+3x^4+4x^3+3$
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 \(\Q(\zeta_{12})\). |
The base change of $A$ to $\F_{5^{3}}$ is 1.125.e 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.