Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 8 x + 41 x^{2} + 144 x^{3} + 533 x^{4} + 1352 x^{5} + 2197 x^{6}$ |
| Frobenius angles: | $\pm0.392572804923$, $\pm0.730542987792$, $\pm0.783731214358$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.8945718272.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4276$ | $5421968$ | $10412286628$ | $23760538551296$ | $50780374288207156$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $22$ | $188$ | $2158$ | $29124$ | $368342$ | $4825916$ | $62775182$ | $815694140$ | $10604729014$ | $137857547868$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=2 x^7+9 x^6+x^5+7 x^4+10 x^3+9 x^2+5 x+6$
- $y^2=2 x^7+9 x^6+x^5+3 x^4+12 x^3+4 x+11$
- $y^2=x^8+3 x^7+12 x^6+10 x^5+12 x^4+8 x^3+3 x^2+8 x+7$
- $y^2=x^8+7 x^7+6 x^6+4 x^5+6 x^4+3 x^3+12 x^2+9 x+12$
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 6.0.8945718272.1. |
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.ai_bp_afo | $2$ | (not in LMFDB) |