Properties

Label 2.3.ae_i
Base field $\F_{3}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
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:  $1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4}$
Frobenius angles:  $\pm0.0540867239847$, $\pm0.445913276015$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\zeta_{8})\)
Galois group:  $C_2^2$
Jacobians:  1

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2$ $68$ $626$ $4624$ $49282$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $0$ $10$ $24$ $54$ $200$ $730$ $2240$ $6494$ $19392$ $59050$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\).
Endomorphism algebra over $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
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.e_i$2$2.9.a_ao
2.3.ae_k$8$(not in LMFDB)
2.3.a_ac$8$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.e_i$2$2.9.a_ao
2.3.ae_k$8$(not in LMFDB)
2.3.a_ac$8$(not in LMFDB)
2.3.a_c$8$(not in LMFDB)
2.3.e_k$8$(not in LMFDB)
2.3.ac_b$24$(not in LMFDB)
2.3.c_b$24$(not in LMFDB)