Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $4$ |
| L-polynomial: | $1 - 2 x^{2} + 38 x^{4} - 50 x^{6} + 625 x^{8}$ |
| Frobenius angles: | $\pm0.173825181470$, $\pm0.291952788861$, $\pm0.708047211139$, $\pm0.826174818530$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 8.0.4868483383296.1 |
| Galois group: | $C_2^2 \wr C_2$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 3$ |
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: | $4$ |
| Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $612$ | $374544$ | $245221668$ | $192012228864$ | $95384984029092$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $22$ | $126$ | $770$ | $3126$ | $15766$ | $78126$ | $390234$ | $1953126$ | $9769222$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^9+x^8+3 x^6+x^5+4 x^4+x^3+4 x^2+x+2$
- $y^2=2 x^9+2 x^8+x^6+2 x^5+3 x^4+2 x^3+3 x^2+2 x+4$
- $y^2=x^{10}+2 x^8+3 x^7+3 x^6+4 x^5+2 x^3+x+4$
- $y^2=2 x^{10}+4 x^8+x^7+x^6+3 x^5+4 x^3+2 x+3$
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 8.0.4868483383296.1. |
| The base change of $A$ to $\F_{5^{2}}$ is 2.25.ac_bm 2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.137904.1$)$ |
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_c_a_bm | $4$ | (not in LMFDB) |