Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 23 x^{2} - 18 x^{3} + 299 x^{4} + 2197 x^{6}$ |
| Frobenius angles: | $\pm0.286919669726$, $\pm0.555271725591$, $\pm0.647776216412$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.5364381632.2 |
| Galois group: | $S_4\times C_2$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2502$ | $6350076$ | $10341999486$ | $23410977591024$ | $51472151345634822$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $216$ | $2144$ | $28700$ | $373364$ | $4822092$ | $62719538$ | $815734172$ | $10604580698$ | $137858998356$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 29 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^8+2 x^7+11 x^6+4 x^5+x^4+3 x^3+4 x^2+12 x+11$
- $y^2=2 x^8+11 x^7+12 x^6+7 x^4+12 x^3+9 x^2+5 x+5$
- $y^2=x^8+8 x^7+6 x^6+11 x^5+4 x^4+8 x^3+3 x^2+x+9$
- $y^2=x^8+4 x^7+5 x^6+12 x^5+6 x^4+5 x^3+7 x^2+9 x+12$
- $y^2=x^8+10 x^7+9 x^6+8 x^5+3 x^4+8 x^3+9 x^2+x+2$
- $y^2=x^8+10 x^7+7 x^6+2 x^5+10 x^4+10 x^3+5 x^2+1$
- $y^2=2 x^8+12 x^7+6 x^6+11 x^5+5 x^4+x^3+3 x^2+2 x+6$
- $y^2=2 x^8+7 x^7+4 x^4+x^3+9 x^2+x+10$
- $y^2=x^8+12 x^7+4 x^6+x^5+3 x^4+8 x^3+6 x^2+7 x+4$
- $y^2=2 x^8+12 x^7+2 x^6+8 x^5+3 x^4+6 x^3+11 x^2+6 x+5$
- $y^2=x^8+3 x^7+7 x^6+5 x^4+8 x^3+3 x^2+2 x+12$
- $y^2=x^8+10 x^7+7 x^6+2 x^5+11 x^3+11 x^2+7 x+1$
- $y^2=x^8+11 x^7+11 x^6+6 x^5+12 x^4+12 x^3+3 x+10$
- $y^2=2 x^8+x^7+9 x^5+7 x^4+4 x^3+6 x^2+11 x+1$
- $y^2=2 x^8+8 x^7+2 x^6+12 x^5+6 x^4+7 x^3+5 x^2+4 x+12$
- $y^2=2 x^8+11 x^7+9 x^6+4 x^4+x^3+8 x^2+2 x+8$
- $y^2=2 x^8+8 x^7+6 x^6+11 x^5+2 x^4+10 x^3+6 x^2+9$
- $y^2=x^8+11 x^7+4 x^6+5 x^5+4 x^4+11 x^3+12 x^2+5 x+9$
- $y^2=x^8+2 x^7+5 x^6+10 x^5+x^3+8 x+6$
- $y^2=x^8+6 x^7+3 x^6+10 x^4+8 x^3+6 x^2+7 x+5$
- $y^2=2 x^8+x^7+3 x^6+7 x^5+x^4+7 x^3+3 x^2+6 x+10$
- $y^2=x^8+3 x^7+10 x^6+9 x^5+9 x^4+9 x^3+11 x^2+x+7$
- $y^2=2 x^8+4 x^7+4 x^6+12 x^5+8 x^4+8 x^3+12 x^2+9 x+10$
- $y^2=2 x^8+8 x^7+8 x^6+6 x^5+9 x^4+2 x^3+8 x^2+2 x+3$
- $y^2=x^8+9 x^7+2 x^6+4 x^5+8 x^4+10 x^3+9 x^2+x+6$
- $y^2=2 x^8+8 x^7+8 x^6+7 x^4+x^3+12 x^2+4 x+2$
- $y^2=x^8+10 x^7+2 x^6+3 x^5+3 x^4+11 x^3+9 x^2+1$
- $y^2=x^8+9 x^7+12 x^6+4 x^5+5 x^4+4 x^3+3 x^2+8 x+12$
- $y^2=2 x^8+6 x^7+9 x^5+3 x^4+12 x^3+5 x^2+8 x+9$
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 6.0.5364381632.2. |
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 |
|---|---|---|
| 3.13.a_x_s | $2$ | (not in LMFDB) |