Properties

 Label 2.3.b_b Base field $\F_{3}$ Dimension $2$ $p$-rank $2$ Ordinary yes Supersingular no Simple yes Geometrically simple yes Primitive yes Principally polarizable yes Contains a Jacobian yes

Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $1 + x + x^{2} + 3 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.327011428181$, $\pm0.798251144367$ Angle rank: $2$ (numerical) Number field: 4.0.16317.1 Galois group: $D_{4}$ Jacobians: 2

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary. $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=2x^5+2x^4+x+2$
• $y^2=2x^6+x^4+2x^3+x^2+x$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $15$ $105$ $945$ $8925$ $49200$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $11$ $35$ $107$ $200$ $719$ $2105$ $6563$ $20195$ $58886$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is 4.0.16317.1.
All geometric endomorphisms are defined over $\F_{3}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.ab_b$2$2.9.b_n