Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 5 x + 28 x^{2} - 65 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.281998603395$, $\pm0.480634277973$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.29189.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $12$ |
| Isomorphism classes: | 12 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $128$ | $34304$ | $5052416$ | $815200256$ | $137849827968$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $201$ | $2298$ | $28545$ | $371269$ | $4828422$ | $62739273$ | $815640609$ | $10604416674$ | $137859728161$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=8 x^6+3 x^5+6 x^3+9 x^2+12 x+6$
- $y^2=5 x^6+4 x^5+11 x^4+11 x^3+2 x^2+10 x+3$
- $y^2=x^6+4 x^5+12 x^4+6 x^3+8 x^2+12 x+8$
- $y^2=7 x^6+6 x^5+3 x^4+x^3+3 x+7$
- $y^2=2 x^5+12 x^4+10 x^3+12 x^2+7$
- $y^2=6 x^6+12 x^4+12 x^3+2 x^2+10 x+2$
- $y^2=12 x^6+2 x^5+5 x^4+6 x^3+2 x^2+x$
- $y^2=6 x^6+4 x^5+5 x^4+12 x^3+x+12$
- $y^2=5 x^5+3 x^4+7 x+5$
- $y^2=6 x^6+4 x^5+9 x^4+4 x^3+8 x^2+8 x+6$
- $y^2=8 x^5+12 x^4+8 x^3+3 x^2+x+2$
- $y^2=10 x^6+x^5+x^3+11 x^2+10 x+4$
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 4.0.29189.1. |
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_bc | $2$ | 2.169.bf_se |