Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 6 x + 11 x^{2} )( 1 - 3 x + 11 x^{2} )$ |
$1 - 9 x + 40 x^{2} - 99 x^{3} + 121 x^{4}$ | |
Frobenius angles: | $\pm0.140218899004$, $\pm0.350615407277$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $2$ |
Isomorphism classes: | 10 |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $54$ | $14580$ | $1844856$ | $216133920$ | $25921530954$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $121$ | $1386$ | $14761$ | $160953$ | $1771378$ | $19494723$ | $214410001$ | $2358099486$ | $25937515801$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+5x^5+9x^4+3x^3+6x^2+7x+10$
- $y^2=10x^6+4x^4+10x^3+5x^2+8x+7$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11}$.
Endomorphism algebra over $\F_{11}$The isogeny class factors as 1.11.ag $\times$ 1.11.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.ad_e | $2$ | 2.121.ab_ci |
2.11.d_e | $2$ | 2.121.ab_ci |
2.11.j_bo | $2$ | 2.121.ab_ci |