Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 8 x + 34 x^{2} + 104 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.551945403333$, $\pm0.895835647611$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.1088.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 10 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $316$ | $29072$ | $4846492$ | $805643264$ | $138159169276$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $22$ | $174$ | $2206$ | $28206$ | $372102$ | $4830942$ | $62723662$ | $815765214$ | $10604502262$ | $137859182734$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=10 x^6+8 x^5+2 x^4+8 x^3+9 x+3$
- $y^2=8 x^6+10 x^5+11 x^4+3 x^3+7 x^2+12$
- $y^2=6 x^5+6 x^4+10 x^3+3 x^2+x+9$
- $y^2=12 x^6+x^5+7 x^4+12 x^3+11 x+9$
- $y^2=10 x^6+3 x^5+2 x^3+6 x+4$
- $y^2=10 x^6+3 x^5+5 x^3+2 x^2+x+6$
- $y^2=8 x^6+6 x^4+6 x^3+11 x^2+6 x+3$
- $y^2=x^6+7 x^5+4 x^4+7 x^3+12 x^2+5 x+6$
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 4.0.1088.2. |
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.ai_bi | $2$ | 2.169.e_ago |