Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 8 x + 40 x^{2} - 104 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.229660330050$, $\pm0.383259645523$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.65792.3 |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $98$ | $31556$ | $5134514$ | $824116496$ | $137825279298$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $186$ | $2334$ | $28854$ | $371206$ | $4825866$ | $62751870$ | $815736798$ | $10604330598$ | $137857406746$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+4x^5+4x^4+x^3+10x^2+4$
- $y^2=2x^6+5x^5+x^4+x^3+8x^2+5x+8$
- $y^2=4x^6+3x^5+10x^4+5x^3+8x^2+5x+11$
- $y^2=11x^6+10x^5+8x^4+9x^3+5x^2+12x+8$
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 4.0.65792.3. |
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.i_bo | $2$ | 2.169.q_ko |