Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 8 x + 37 x^{2} + 104 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.578661031392$, $\pm0.832546644796$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.16025.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 10 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $319$ | $30305$ | $4690576$ | $815356025$ | $137890685119$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $22$ | $180$ | $2134$ | $28548$ | $371382$ | $4831590$ | $62719294$ | $815780868$ | $10604667502$ | $137857522900$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=x^6+9 x^5+7 x^4+4 x^3+7 x^2+x+6$
- $y^2=3 x^6+10 x^5+7 x^2+2 x+3$
- $y^2=10 x^6+5 x^5+3 x^4+7 x^2+2 x+9$
- $y^2=4 x^6+4 x^5+3 x^4+12 x^2+4 x+3$
- $y^2=7 x^6+12 x^5+4 x^4+11 x^3+3 x^2+5 x+11$
- $y^2=2 x^6+3 x^5+11 x^4+5 x^3+5 x^2+7 x+9$
- $y^2=x^6+9 x^5+2 x^4+9 x^3+6 x^2+9 x+1$
- $y^2=12 x^6+x^5+9 x^3+7 x^2+11 x+3$
- $y^2=9 x^6+2 x^5+3 x^4+6 x^3+4 x^2+6$
- $y^2=x^6+4 x^5+9 x^4+6 x^3+4 x^2+11 x+1$
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 4.0.16025.1. |
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.ai_bl | $2$ | 2.169.k_br |