Properties

Label 2.4.ad_h
Base Field $\F_{2^{2}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 3 x + 7 x^{2} - 12 x^{3} + 16 x^{4}$
Frobenius angles:  $\pm0.190783854037$, $\pm0.524117187371$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{13})\)
Galois group:  $C_2^2$
Jacobians:  4

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 Jacobians of 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 9 351 4212 63531 1141299 17740944 268402689 4264772499 68719584492 1099012730751

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 22 65 250 1112 4327 16382 65074 262145 1048102

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{13})\).
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.el 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-39}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{12}}$.
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
2.4.d_h$2$2.16.f_j
2.4.a_af$3$2.64.a_el
2.4.d_h$3$2.64.a_el
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.d_h$2$2.16.f_j
2.4.a_af$3$2.64.a_el
2.4.d_h$3$2.64.a_el
2.4.a_f$12$(not in LMFDB)