Invariants
| Base field: | $\F_{5^{4}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 43 x + 625 x^{2}$ |
| Frobenius angles: | $\pm0.170463428383$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-651}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $583$ | $390027$ | $244141744$ | $152588313075$ | $95367449107183$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $583$ | $390027$ | $244141744$ | $152588313075$ | $95367449107183$ | $59604645262421952$ | $37252902994644889519$ | $23283064365513675514275$ | $14551915228366034356399408$ | $9094947017729168033400237627$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which 0 are hyperelliptic):
- $y^2=x^3+a^{211} x+a^{212}$
- $y^2=x^3+a^{423} x+a^{424}$
- $y^2=x^3+a^{235} x+a^{236}$
- $y^2=x^3+a^{463} x+a^{463}$
- $y^2=x^3+a^{543} x+a^{544}$
- $y^2=x^3+a^{467} x+a^{467}$
- $y^2=x^3+a^{343} x+a^{343}$
- $y^2=x^3+a^{443} x+a^{443}$
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 \(\Q(\sqrt{-651}) \). |
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.br | $2$ | (not in LMFDB) |