Properties

Label 2.4.a_b
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:  $1 + x^{2} + 16 x^{4}$
Frobenius angles:  $\pm0.269946543837$, $\pm0.730053456163$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(i, \sqrt{7})\)
Galois group:  $C_2^2$
Jacobians:  6

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 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 18 324 4050 82944 1049778 16402500 268410258 4236447744 68719950450 1102033849284

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 19 65 319 1025 4003 16385 64639 262145 1050979

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{7})\).
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.b 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{4}}$.

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.ag_r$4$2.256.ck_cer
2.4.a_ab$4$2.256.ck_cer
2.4.g_r$4$2.256.ck_cer
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.ag_r$4$2.256.ck_cer
2.4.a_ab$4$2.256.ck_cer
2.4.g_r$4$2.256.ck_cer
2.4.ad_f$12$(not in LMFDB)
2.4.d_f$12$(not in LMFDB)