Properties

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

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $1 + 9 x^{4}$
Frobenius angles:  $\pm0.250000000000$, $\pm0.750000000000$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(i, \sqrt{6})\)
Galois group:  $C_2^2$
Jacobians:  2

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

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $10$ $100$ $730$ $10000$ $59050$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $10$ $28$ $118$ $244$ $730$ $2188$ $6238$ $19684$ $59050$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{6})\).
Endomorphism algebra over $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{4}}$ is 1.81.s 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 $\F_{3^{4}}$.
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_ag$8$(not in LMFDB)
2.3.a_g$8$(not in LMFDB)
2.3.ag_p$24$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.a_ag$8$(not in LMFDB)
2.3.a_g$8$(not in LMFDB)
2.3.ag_p$24$(not in LMFDB)
2.3.ad_g$24$(not in LMFDB)
2.3.a_ad$24$(not in LMFDB)
2.3.a_d$24$(not in LMFDB)
2.3.d_g$24$(not in LMFDB)
2.3.g_p$24$(not in LMFDB)