Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 5 x + 15 x^{2} - 55 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.123520620106$, $\pm0.554980822636$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.755621.1 |
Galois group: | $D_{4}$ |
Jacobians: | $7$ |
Isomorphism classes: | 7 |
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: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $77$ | $15169$ | $1688687$ | $211592381$ | $26082870352$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $127$ | $1267$ | $14451$ | $161952$ | $1774747$ | $19486957$ | $214382643$ | $2358135997$ | $25937612502$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=5x^6+6x^5+8x^4+2x^3+9x^2+8x+6$
- $y^2=6x^6+8x^4+9x^3+5x^2+8x+9$
- $y^2=9x^6+8x^5+7x^4+3x^3+5x^2+8$
- $y^2=6x^6+x^5+4x^4+5x^3+6x^2+2x+9$
- $y^2=6x^6+7x^5+3x^4+6x^3+x^2+7$
- $y^2=6x^6+4x^5+3x^3+2x^2+10x+4$
- $y^2=6x^6+5x^5+5x^4+9x^3+8x^2+2x$
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 4.0.755621.1. |
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 |
---|---|---|
2.11.f_p | $2$ | 2.121.f_adf |