Properties

Label 2.4.ad_f
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:  $1 - 3 x + 5 x^{2} - 12 x^{3} + 16 x^{4}$
Frobenius angles:  $\pm0.103279877171$, $\pm0.563386789496$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{-7})\)
Galois group:  $C_2^2$
Jacobians:  2

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 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 7 259 3136 58275 1109227 17172736 267001147 4324529475 69244764736 1100771362579

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 18 47 226 1082 4191 16298 65986 264143 1049778

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-7})\).
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{6}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.

SubfieldPrimitive Model
$\F_{2}$2.2.ab_ab
$\F_{2}$2.2.b_ab

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.d_f$2$2.16.b_ap
2.4.g_r$3$2.64.as_ib
2.4.ag_r$6$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.d_f$2$2.16.b_ap
2.4.g_r$3$2.64.as_ib
2.4.ag_r$6$(not in LMFDB)
2.4.a_ab$6$(not in LMFDB)
2.4.a_b$12$(not in LMFDB)