Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 5 x + 33 x^{2} + 102 x^{3} + 363 x^{4} + 605 x^{5} + 1331 x^{6}$ |
| Frobenius angles: | $\pm0.444956768062$, $\pm0.573354514772$, $\pm0.745671237289$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.3245597440.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2440$ | $2479040$ | $2248303840$ | $3118790978560$ | $4166738882251000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $17$ | $163$ | $1268$ | $14551$ | $160647$ | $1773196$ | $19493421$ | $214332207$ | $2357995436$ | $25937201763$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=5 x^8+4 x^7+9 x^6+7 x^5+7 x^4+9 x^3+8 x^2+10 x$
- $y^2=10 x^8+x^7+3 x^5+4 x^4+6 x^3+x^2+5 x+9$
- $y^2=7 x^8+7 x^7+4 x^6+3 x^5+5 x^4+6 x^3+x^2+7 x+9$
- $y^2=3 x^8+x^5+9 x^4+x^3+8 x^2+3 x+5$
- $y^2=x^8+10 x^7+3 x^6+9 x^5+10 x^4+x^3+8 x^2+7 x+9$
- $y^2=5 x^8+6 x^7+4 x^5+3 x^4+4 x^3+4 x^2+9 x+1$
- $y^2=7 x^8+4 x^7+5 x^6+10 x^5+6 x^4+9 x^3+2 x^2+9 x+4$
- $y^2=2 x^8+4 x^7+x^6+6 x^5+8 x^4+9 x^3+9 x^2+9 x$
- $y^2=2 x^8+2 x^7+2 x^6+8 x^5+3 x^4+2 x^3+x^2+7 x+9$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is 6.0.3245597440.1. |
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 |
|---|---|---|
| 3.11.af_bh_ady | $2$ | (not in LMFDB) |