# Properties

 Label 2.2.ac_c Base Field $\F_{2}$ Dimension $2$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4}$ Frobenius angles: $\pm0.0833333333333$, $\pm0.583333333333$ Angle rank: $0$ (numerical) Number field: $$\Q(\zeta_{12})$$ 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:

• $y^2+y=x^5+x^3+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1 13 25 169 1321 4225 14449 74529 297025 1047553

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 1 5 1 9 41 65 113 289 577 1025

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\zeta_{12})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey 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 $\F_{2^{12}}$.
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.a_ae and its endomorphism algebra is $$\Q(\zeta_{12})$$.
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$

## 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.c_c $2$ 2.4.a_ae 2.2.e_i $3$ 2.8.ai_bg 2.2.ae_i $6$ 2.64.a_ey
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.2.c_c $2$ 2.4.a_ae 2.2.e_i $3$ 2.8.ai_bg 2.2.ae_i $6$ 2.64.a_ey 2.2.a_a $6$ 2.64.a_ey 2.2.a_ac $8$ 2.256.bg_bdo 2.2.a_c $8$ 2.256.bg_bdo 2.2.ac_e $24$ (not in LMFDB) 2.2.a_ae $24$ (not in LMFDB) 2.2.a_e $24$ (not in LMFDB) 2.2.c_e $24$ (not in LMFDB)