Properties

Label 2.3.ag_p
Base Field $\F_{3}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )^{2}$
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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1 49 784 8281 73441 614656 5148361 44129449 387459856 3458263249

Point counts of the (virtual) curve

$r$ 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

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)