Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 6 x + 18 x^{2} - 66 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.0290991158233$, $\pm0.529099115823$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{13})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 3 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $68$ | $14416$ | $1655732$ | $207821056$ | $25878546548$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $122$ | $1242$ | $14190$ | $160686$ | $1771562$ | $19474818$ | $214315294$ | $2357952822$ | $25937424602$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+6x^5+x^4+x^2+5x+8$
- $y^2=9x^5+6x^4+3x^3+5x^2+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{4}}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{13})\). |
| The base change of $A$ to $\F_{11^{4}}$ is 1.14641.ais 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$ |
- Endomorphism algebra over $\F_{11^{2}}$
The base change of $A$ to $\F_{11^{2}}$ is the simple isogeny class 2.121.a_ais and its endomorphism algebra is \(\Q(i, \sqrt{13})\).
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.g_s | $2$ | 2.121.a_ais |
| 2.11.a_ae | $8$ | (not in LMFDB) |
| 2.11.a_e | $8$ | (not in LMFDB) |