Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 4 x + 39 x^{2} + 98 x^{3} + 507 x^{4} + 676 x^{5} + 2197 x^{6}$ |
| Frobenius angles: | $\pm0.451873281164$, $\pm0.569952055271$, $\pm0.662026604255$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.31747253184.1 |
| Galois group: | $S_4\times C_2$ |
| Cyclic group of points: | yes |
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: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3522$ | $6924252$ | $10085954922$ | $23021116018416$ | $51314792969574642$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $232$ | $2088$ | $28220$ | $372228$ | $4826236$ | $62752974$ | $815755292$ | $10604282886$ | $137858373892$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=2 x^8+9 x^7+x^6+6 x^5+x^4+12 x^3+5 x^2+6 x+1$
- $y^2=x^8+2 x^7+6 x^6+x^5+2 x^4+7 x^3+x^2+7 x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$| The endomorphism algebra of this simple isogeny class is 6.0.31747253184.1. |
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 |
|---|---|---|
| 3.13.ae_bn_adu | $2$ | (not in LMFDB) |