Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 6 x + 11 x^{2} )( 1 + x + 11 x^{2} )$ |
$1 - 5 x + 16 x^{2} - 55 x^{3} + 121 x^{4}$ | |
Frobenius angles: | $\pm0.140218899004$, $\pm0.548170674452$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $4$ |
Isomorphism classes: | 25 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $78$ | $15444$ | $1708200$ | $212138784$ | $26103086178$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $129$ | $1282$ | $14489$ | $162077$ | $1775538$ | $19488287$ | $214375729$ | $2358108742$ | $25937540529$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=10x^6+6x^3+5x^2+7x+4$
- $y^2=10x^6+3x^5+3x^4+5x^3+7x^2+8x+4$
- $y^2=7x^6+3x^5+3x^4+x^3+2x^2+6x+2$
- $y^2=7x^6+9x^5+6x^4+5x^3+5x^2+3x+8$
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.b 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.ah_bc | $2$ | 2.121.h_aca |
2.11.f_q | $2$ | 2.121.h_aca |
2.11.h_bc | $2$ | 2.121.h_aca |