Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 5 x + 18 x^{2} + 65 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.443425678274$, $\pm0.835993991057$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.2066364.2 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $258$ | $30444$ | $4941216$ | $812976576$ | $137126902218$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $19$ | $181$ | $2248$ | $28465$ | $369319$ | $4833322$ | $62749363$ | $815754049$ | $10604278024$ | $137858023261$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=x^6+11 x^5+10 x^4+6 x^3+2 x^2+2 x+3$
- $y^2=9 x^6+6 x^5+x^3+4 x^2+2$
- $y^2=10 x^6+9 x^5+3 x^4+12 x^3+7 x^2+7 x+5$
- $y^2=3 x^6+5 x^5+x^4+4 x^3+12 x^2+7 x+10$
- $y^2=11 x^6+11 x^5+6 x^4+11 x^3+9 x^2+6 x+9$
- $y^2=4 x^6+8 x^5+8 x^4+x^3+x^2+8 x+5$
- $y^2=6 x^5+5 x^4+10 x^3+6 x^2+9$
- $y^2=6 x^6+12 x^5+10 x^4+x^3+2 x^2+4$
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 4.0.2066364.2. |
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.af_s | $2$ | 2.169.l_m |