# Properties

 Label 2.2.a_c Base field $\F_{2}$ Dimension $2$ $p$-rank $0$ Ordinary No Supersingular Yes Simple Yes Geometrically simple No Primitive Yes Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $1 + 2 x^{2} + 4 x^{4}$ Frobenius angles: $\pm0.333333333333$, $\pm0.666666666667$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Galois group: $C_2^2$ Jacobians: 1

This isogeny class is simple but not geometrically simple.

## Newton polygon

This isogeny class is supersingular. $p$-rank: $0$ Slopes: $[1/2, 1/2, 1/2, 1/2]$

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

• $y^2+y=x^5+x^4+x^3$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 7 49 49 441 1057 2401 16513 74529 261121 1117249

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 9 9 25 33 33 129 289 513 1089

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{2}, \sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{6}}$ is 1.64.aq 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
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 1.4.c 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is the simple isogeny class 2.8.a_aq and its endomorphism algebra is the quaternion algebra over $$\Q(\sqrt{2})$$ ramified at both real infinite places.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.2.a_ae $3$ 2.8.a_aq 2.2.a_ac $4$ 2.16.i_bw 2.2.a_ae $6$ 2.64.abg_ou
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.2.a_ae $3$ 2.8.a_aq 2.2.a_ac $4$ 2.16.i_bw 2.2.a_ae $6$ 2.64.abg_ou 2.2.ac_c $8$ 2.256.bg_bdo 2.2.c_c $8$ 2.256.bg_bdo 2.2.a_e $12$ (not in LMFDB) 2.2.ae_i $24$ (not in LMFDB) 2.2.ac_e $24$ (not in LMFDB) 2.2.a_a $24$ (not in LMFDB) 2.2.c_e $24$ (not in LMFDB) 2.2.e_i $24$ (not in LMFDB)