Invariants
Base field: | $\F_{23}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 15 x + 98 x^{2} - 345 x^{3} + 529 x^{4}$ |
Frobenius angles: | $\pm0.0252283536038$, $\pm0.308104979730$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-3}, \sqrt{17})\) |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $268$ | $264784$ | $148014256$ | $78250026816$ | $41392253570068$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $9$ | $501$ | $12168$ | $279625$ | $6431019$ | $147992622$ | $3404623413$ | $78310472689$ | $1801152661464$ | $41426514634461$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=17x^6+19x^5+8x^4+9x^3+10x^2+10x+3$
- $y^2=19x^6+5x^5+5x^4+14x^3+12x^2+17x+19$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23^{6}}$.
Endomorphism algebra over $\F_{23}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{17})\). |
The base change of $A$ to $\F_{23^{6}}$ is 1.148035889.abgac 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$ |
- Endomorphism algebra over $\F_{23^{2}}$
The base change of $A$ to $\F_{23^{2}}$ is the simple isogeny class 2.529.abd_ma and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{17})\). - Endomorphism algebra over $\F_{23^{3}}$
The base change of $A$ to $\F_{23^{3}}$ is the simple isogeny class 2.12167.a_abgac and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{17})\).
Base change
This is a primitive isogeny class.