Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 8 x^{2} - 20 x^{3} + 25 x^{4}$ |
Frobenius angles: | $\pm0.0320471084245$, $\pm0.532047108424$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(i, \sqrt{6})\) |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $10$ | $580$ | $12490$ | $336400$ | $9629050$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $26$ | $98$ | $534$ | $3082$ | $15626$ | $77338$ | $388894$ | $1953602$ | $9765626$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+4x^5+x^4+x^2+x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{4}}$.
Endomorphism algebra over $\F_{5}$The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{6})\). |
The base change of $A$ to $\F_{5^{4}}$ is 1.625.abu 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$ |
- Endomorphism algebra over $\F_{5^{2}}$
The base change of $A$ to $\F_{5^{2}}$ is the simple isogeny class 2.25.a_abu and its endomorphism algebra is \(\Q(i, \sqrt{6})\).
Base change
This is a primitive isogeny class.