Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 5 x + 24 x^{2} - 65 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.232448793773$, $\pm0.516440416321$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.326700.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $12$ |
| Isomorphism classes: | 12 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $124$ | $32736$ | $4918336$ | $815650176$ | $138370094764$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $193$ | $2238$ | $28561$ | $372669$ | $4832422$ | $62736753$ | $815626753$ | $10604394294$ | $137858748553$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=6 x^6+12 x^5+6 x^4+4 x^3+8 x^2+8 x+10$
- $y^2=2 x^6+9 x^5+10 x^4+8 x^3+6 x+4$
- $y^2=4 x^6+10 x^5+3 x^4+3 x^3+3 x^2+9 x$
- $y^2=7 x^6+8 x^5+7 x^4+9 x^3+10 x^2+8$
- $y^2=x^6+3 x^5+11 x^4+4 x^3+9 x^2+12 x+6$
- $y^2=12 x^6+3 x^5+10 x^4+3 x^3+9 x^2+x+11$
- $y^2=8 x^6+8 x^5+12 x^4+5 x^3+6 x+7$
- $y^2=5 x^6+3 x^5+12 x^4+6 x^3+4 x+3$
- $y^2=6 x^6+6 x^5+11 x^2+8 x+1$
- $y^2=6 x^6+8 x^5+x^4+x^3+x^2+5 x$
- $y^2=x^5+3 x^4+12 x^3+8 x^2+2 x+11$
- $y^2=12 x^6+x^5+2 x^4+5 x^3+5 x^2+8 x+3$
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 4.0.326700.1. |
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.f_y | $2$ | 2.169.x_ke |