Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 2 x^{2} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.235511365082$, $\pm0.764488634918$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-5}, \sqrt{6})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $22$ |
| Isomorphism classes: | 76 |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $120$ | $14400$ | $1772280$ | $221414400$ | $25937283000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $118$ | $1332$ | $15118$ | $161052$ | $1772998$ | $19487172$ | $214304158$ | $2357947692$ | $25937141398$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 22 curves (of which all are hyperelliptic):
- $y^2=10 x^5+10 x^3+5 x^2+9 x$
- $y^2=9 x^5+9 x^3+10 x^2+7 x$
- $y^2=4 x^6+5 x^5+x^4+x^3+9 x^2+9 x+3$
- $y^2=6 x^6+8 x^5+9 x^4+6 x^3+2 x^2+6 x+7$
- $y^2=x^6+5 x^5+7 x^4+x^3+4 x^2+x+3$
- $y^2=8 x^6+9 x^5+10 x^4+4 x^3+2 x^2+2 x+7$
- $y^2=8 x^5+8 x^4+2 x^3+6 x^2+8 x+1$
- $y^2=2 x^5+7 x^4+x^3+2 x^2+7 x+3$
- $y^2=4 x^6+8 x^5+7 x^4+5 x^3+6 x^2+9 x+4$
- $y^2=4 x^6+8 x^5+5 x^4+8 x^3+4 x^2+2 x+4$
- $y^2=8 x^6+5 x^5+10 x^4+5 x^3+8 x^2+4 x+8$
- $y^2=6 x^6+10 x^5+9 x^4+4 x^3+7 x^2+4 x+9$
- $y^2=x^6+9 x^5+7 x^4+8 x^3+3 x^2+8 x+7$
- $y^2=7 x^6+2 x^5+8 x^4+4 x^3+3 x^2+8 x+9$
- $y^2=3 x^6+4 x^5+5 x^4+8 x^3+6 x^2+5 x+7$
- $y^2=x^6+7 x^4+x^3+6 x^2+8 x+7$
- $y^2=7 x^6+4 x^5+5 x^4+9 x^3+8 x^2+7 x$
- $y^2=3 x^6+8 x^5+10 x^4+7 x^3+5 x^2+3 x$
- $y^2=3 x^6+10 x^5+7 x^4+x^3+3 x^2+7 x+4$
- $y^2=5 x^6+10 x^5+3 x^4+3 x^3+10 x^2+6 x+3$
- $y^2=9 x^6+5 x^5+6 x^4+2 x^3+8 x^2+8 x+1$
- $y^2=7 x^6+10 x^5+x^4+4 x^3+5 x^2+5 x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{2}}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-5}, \sqrt{6})\). |
| The base change of $A$ to $\F_{11^{2}}$ is 1.121.ac 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-30}) \)$)$ |
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 |
|---|---|---|
| 2.11.a_c | $4$ | (not in LMFDB) |