Properties

 Label 2.3.ag_p Base field $\F_{3}$ Dimension $2$ $p$-rank $0$ Ordinary no Supersingular yes Simple no Geometrically simple no Primitive yes Principally polarizable yes Contains a Jacobian no

Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $( 1 - 3 x + 3 x^{2} )^{2}$ $1 - 6 x + 15 x^{2} - 18 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$ Angle rank: $0$ (numerical) Jacobians: 0

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $49$ $784$ $8281$ $73441$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $4$ $28$ $100$ $298$ $838$ $2350$ $6724$ $19684$ $58564$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.a_ad$2$2.9.ag_bb
2.3.g_p$2$2.9.ag_bb
2.3.ad_g$3$2.27.a_cc
2.3.a_ad$3$2.27.a_cc
2.3.a_g$3$2.27.a_cc
2.3.d_g$3$2.27.a_cc
2.3.g_p$3$2.27.a_cc
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.a_ad$2$2.9.ag_bb
2.3.g_p$2$2.9.ag_bb
2.3.ad_g$3$2.27.a_cc
2.3.a_ad$3$2.27.a_cc
2.3.a_g$3$2.27.a_cc
2.3.d_g$3$2.27.a_cc
2.3.g_p$3$2.27.a_cc
2.3.a_d$4$2.81.s_jj
2.3.a_ad$6$2.729.ee_gmg
2.3.a_ag$12$(not in LMFDB)
2.3.a_a$24$(not in LMFDB)