Properties

Label 2.2.d_f
Base field $\F_{2}$
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:  $\F_{2}$
Dimension:  $2$
L-polynomial:  $1 + 3 x + 5 x^{2} + 6 x^{3} + 4 x^{4}$
Frobenius angles:  $\pm0.543118021706$, $\pm0.876451355039$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{5})\)
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $19$ $19$ $76$ $171$ $1159$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $6$ $6$ $9$ $10$ $36$ $87$ $90$ $274$ $513$ $1086$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{5})\).
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.l 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{6}}$.
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
2.2.ad_f$2$2.4.b_ad
2.2.ad_f$3$2.8.a_l
2.2.a_ab$3$2.8.a_l
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.2.ad_f$2$2.4.b_ad
2.2.ad_f$3$2.8.a_l
2.2.a_ab$3$2.8.a_l
2.2.a_b$12$(not in LMFDB)