Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 2 x + 7 x^{2} + 36 x^{3} + 91 x^{4} + 338 x^{5} + 2197 x^{6}$ |
| Frobenius angles: | $\pm0.255153203486$, $\pm0.521589444186$, $\pm0.860237082472$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.1079160000.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2672$ | $5130240$ | $10976183216$ | $23357777510400$ | $51252557210997872$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $16$ | $180$ | $2272$ | $28636$ | $371776$ | $4834260$ | $62713072$ | $815678396$ | $10604352496$ | $137859424500$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 332 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^7+11 x^6+3 x^5+7 x^4+8 x^3+12 x^2+4 x$
- $y^2=x^7+6 x^6+9 x^5+2 x^4+8 x^3+11 x^2+3 x$
- $y^2=2 x^7+7 x^6+11 x^5+4 x^4+9 x^3+4 x^2+11 x+5$
- $y^2=x^7+8 x^6+7 x^5+3 x^4+11 x^3+x^2+4 x+11$
- $y^2=2 x^8+4 x^7+5 x^6+8 x^5+11 x^3+3 x^2+4 x+8$
- $y^2=x^8+5 x^7+4 x^6+12 x^5+2 x^4+5 x^3+8 x+3$
- $y^2=2 x^7+8 x^6+8 x^5+5 x^3+4 x^2+2 x+10$
- $y^2=x^7+x^6+9 x^5+3 x^4+5 x^3+2 x^2+8 x+10$
- $y^2=2 x^7+4 x^6+12 x^5+11 x^4+7 x^3+x^2+5 x+10$
- $y^2=2 x^7+10 x^6+3 x^5+6 x^4+12 x^3+7 x^2+x+8$
- $y^2=x^7+5 x^6+4 x^5+5 x^4+7 x^3+3 x^2+4 x+12$
- $y^2=2 x^7+8 x^6+x^5+4 x^4+6 x^3+11 x^2+6 x+11$
- $y^2=2 x^7+4 x^6+4 x^4+3 x^3+5 x^2+2 x+7$
- $y^2=2 x^7+8 x^6+4 x^5+3 x^4+10 x^3+x^2+8 x+6$
- $y^2=2 x^7+2 x^6+9 x^5+12 x^4+12 x^3+7 x^2+6 x+5$
- $y^2=x^7+2 x^6+7 x^5+3 x^4+10 x^3+9 x^2+11 x+4$
- $y^2=x^8+2 x^7+2 x^6+8 x^5+3 x^4+5 x^3+10 x^2+2 x+6$
- $y^2=x^8+x^7+12 x^6+8 x^5+11 x^4+11 x^3+x^2+2 x+10$
- $y^2=x^8+x^7+x^6+10 x^5+7 x^4+12 x^3+6 x^2+2 x+10$
- $y^2=2 x^8+4 x^7+10 x^6+6 x^5+5 x^4+11 x^3+10 x^2+2 x+5$
- and 312 more
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 6.0.1079160000.1. |
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.ac_h_abk | $2$ | (not in LMFDB) |