Properties

Label 2.3.a_ag
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:  $( 1 - 3 x^{2} )^{2}$
Frobenius angles:  $0$, $0$, $1$, $1$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{3}) \)
Galois group:  $C_2$
Jacobians:  0

This isogeny class is simple but not geometrically 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
$A(\F_{q^r})$ 4 16 676 4096 58564 456976 4778596 40960000 387381124 3429742096

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 -2 28 46 244 622 2188 6238 19684 58078

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q(\sqrt{3}) \) ramified at both real infinite places.
Endomorphism algebra over $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.ag 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 $\F_{3^{2}}$.

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_d$3$2.27.a_acc
2.3.a_g$4$2.81.abk_ss
2.3.a_a$8$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.3.a_d$3$2.27.a_acc
2.3.a_g$4$2.81.abk_ss
2.3.a_a$8$(not in LMFDB)
2.3.ag_p$12$(not in LMFDB)
2.3.ad_g$12$(not in LMFDB)
2.3.a_ad$12$(not in LMFDB)
2.3.d_g$12$(not in LMFDB)
2.3.g_p$12$(not in LMFDB)