Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 + 8 x + 36 x^{2} + 88 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.627463458247$, $\pm0.803936164834$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.35072.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $254$ | $15748$ | $1658366$ | $217259408$ | $25925566894$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $20$ | $130$ | $1244$ | $14838$ | $160980$ | $1771858$ | $19480796$ | $214381854$ | $2357976116$ | $25936895490$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+x^5+10x^4+6x^3+6x^2+5$
- $y^2=4x^6+7x^5+3x^4+10x^3+x^2+8x+4$
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 4.0.35072.1. |
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.ai_bk | $2$ | 2.121.i_fa |