Invariants
| Base field: | $\F_{89}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 89 x^{2} )( 1 + x + 89 x^{2} )$ |
| $1 + x + 178 x^{2} + 89 x^{3} + 7921 x^{4}$ | |
| Frobenius angles: | $\pm0.5$, $\pm0.516878298350$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $0$ |
| Cyclic group of points: | yes |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $8190$ | $65601900$ | $496795178880$ | $3934623524832000$ | $31181938618486423950$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $91$ | $8277$ | $704704$ | $62710913$ | $5584098611$ | $496984040082$ | $44231330071019$ | $3936588565854913$ | $350356404251229376$ | $31181719950768837477$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{89^{2}}$.
Endomorphism algebra over $\F_{89}$| The isogeny class factors as 1.89.a $\times$ 1.89.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{89^{2}}$ is 1.7921.gv $\times$ 1.7921.gw. The endomorphism algebra for each factor is:
|
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 |
|---|---|---|
| 2.89.ab_gw | $2$ | (not in LMFDB) |