Invariants
| Base field: | $\F_{5^{4}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 24 x + 625 x^{2}$ |
| Frobenius angles: | $\pm0.340636655477$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-481}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $16$ |
| Isomorphism classes: | 16 |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $1$ |
| Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $602$ | $391300$ | $244171802$ | $152588217600$ | $95367420003002$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $602$ | $391300$ | $244171802$ | $152588217600$ | $95367420003002$ | $59604644291728900$ | $37252902980284774202$ | $23283064365585226675200$ | $14551915228374319116485402$ | $9094947017729337679721306500$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which 0 are hyperelliptic):
- $y^2=x^3+a^{246} x+a^{247}$
- $y^2=x^3+a^{114} x+a^{114}$
- $y^2=x^3+a^{550} x+a^{551}$
- $y^2=x^3+a^{522} x+a^{522}$
- $y^2=x^3+a^{570} x+a^{570}$
- $y^2=x^3+a^{378} x+a^{379}$
- $y^2=x^3+a^{10} x+a^{11}$
- $y^2=x^3+a^{346} x+a^{346}$
- $y^2=x^3+a^{354} x+a^{354}$
- $y^2=x^3+a^{538} x+a^{538}$
- $y^2=x^3+a^{598} x+a^{599}$
- $y^2=x^3+a^{202} x+a^{203}$
- $y^2=x^3+a^{194} x+a^{194}$
- $y^2=x^3+a^{486} x+a^{487}$
- $y^2=x^3+a^{42} x+a^{43}$
- $y^2=x^3+a^{482} x+a^{482}$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{4}}$.
Endomorphism algebra over $\F_{5^{4}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-481}) \). |
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 |
|---|---|---|
| 1.625.y | $2$ | (not in LMFDB) |