Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 6 x + 28 x^{2} + 78 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.515643414148$, $\pm0.786273092271$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.818496.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
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: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $282$ | $32148$ | $4713066$ | $814887504$ | $137526944682$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $190$ | $2144$ | $28534$ | $370400$ | $4833790$ | $62742980$ | $815642974$ | $10604787572$ | $137858471950$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=7 x^6+5 x^5+5 x^4+7 x^3+2 x^2+x+3$
- $y^2=3 x^6+6 x^5+2 x^4+7 x^3+5 x^2+7 x+10$
- $y^2=7 x^6+x^5+8 x^4+3 x^3+5 x^2+9 x+3$
- $y^2=3 x^6+3 x^5+6 x^4+7 x^3+6 x^2+5 x+3$
- $y^2=7 x^5+x^4+3 x^3+6 x+1$
- $y^2=10 x^6+2 x^5+5 x^4+4 x^3+4 x^2+7 x+4$
- $y^2=12 x^6+2 x^5+6 x^4+7 x^3+x^2+9$
- $y^2=2 x^6+6 x^5+7 x^4+8 x^3+5 x^2+2 x+3$
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 4.0.818496.1. |
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 |
|---|---|---|
| 2.13.ag_bc | $2$ | 2.169.u_he |