# Properties

 Label 2.2.ad_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

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4}$ Frobenius angles: $\pm0.123548644961$, $\pm0.456881978294$ 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:

• $y^2+(x^3+x+1)y=x^6+x^5+x^4+x^3+x^2+x+1$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $19$ $76$ $171$ $961$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $0$ $6$ $9$ $10$ $30$ $87$ $168$ $274$ $513$ $1086$

## 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
• Endomorphism algebra over $\F_{2^{2}}$  The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 2.4.b_ad and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is the simple isogeny class 2.8.a_l and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{5})$$.

## 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.d_f$2$2.4.b_ad
2.2.a_ab$3$2.8.a_l
2.2.d_f$3$2.8.a_l
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.2.d_f$2$2.4.b_ad
2.2.a_ab$3$2.8.a_l
2.2.d_f$3$2.8.a_l
2.2.a_ab$6$2.64.w_jp
2.2.a_b$12$(not in LMFDB)