Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 7 x + 47 x^{2} - 174 x^{3} + 611 x^{4} - 1183 x^{5} + 2197 x^{6}$ |
| Frobenius angles: | $\pm0.247347272016$, $\pm0.388018503618$, $\pm0.527676267490$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.1219196096.1 |
| Galois group: | $S_4\times C_2$ |
| 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: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1492$ | $6296240$ | $11212081600$ | $23268888243200$ | $51215214169854052$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $215$ | $2320$ | $28527$ | $371507$ | $4828580$ | $62738879$ | $815683967$ | $10604493520$ | $137858218575$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^7+7 x^6+4 x^5+3 x^4+8 x^3+10 x^2+8 x+6$
- $y^2=x^7+11 x^6+4 x^5+7 x^4+8 x^3+12 x^2+9 x+2$
- $y^2=x^7+10 x^6+5 x^5+8 x^4+10 x^2+9 x+11$
- $y^2=x^7+9 x^6+3 x^4+10 x^3+5 x^2+7 x+12$
- $y^2=x^7+7 x^6+12 x^5+5 x^4+9 x^3+2 x^2+12 x+6$
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 6.0.1219196096.1. |
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 |
|---|---|---|
| 3.13.h_bv_gs | $2$ | (not in LMFDB) |