Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $4$ |
| L-polynomial: | $1 + 14 x^{2} + 94 x^{4} + 350 x^{6} + 625 x^{8}$ |
| Frobenius angles: | $\pm0.329028055852$, $\pm0.437386669224$, $\pm0.562613330776$, $\pm0.670971944148$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 8.0.206479360000.2 |
| Galois group: | $D_4\times C_2$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $4$ |
| Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1084$ | $1175056$ | $241741756$ | $148921897216$ | $95381250536764$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $54$ | $126$ | $610$ | $3126$ | $15318$ | $78126$ | $391642$ | $1953126$ | $9768454$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains no Jacobian of a hyperelliptic curve, but it is unknown whether it contains a Jacobian of a non-hyperelliptic curve.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5}$| The endomorphism algebra of this simple isogeny class is 8.0.206479360000.2. |
| The base change of $A$ to $\F_{5^{2}}$ is 2.25.o_dq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-34 +2 \sqrt{5}})\)$)$ |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 4.5.a_ao_a_dq | $4$ | (not in LMFDB) |
| 4.5.ac_c_ag_o | $8$ | (not in LMFDB) |
| 4.5.c_c_g_o | $8$ | (not in LMFDB) |