# Properties

 Label 2.4.a_ae Base Field $\F_{2^{2}}$ Dimension $2$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{2}}$ Dimension: $2$ L-polynomial: $1 - 4 x^{2} + 16 x^{4}$ Frobenius angles: $\pm0.166666666667$, $\pm0.833333333333$ 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:

• $y^2+y=x^5+x^3$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 13 169 4225 74529 1047553 17850625 268419073 4328718849 68720001025 1097367287809

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 9 65 289 1025 4353 16385 66049 262145 1046529

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\zeta_{12})$$.
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 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 $\F_{2^{4}}$  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 isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.

 Subfield Primitive Model $\F_{2}$ 2.2.ac_c $\F_{2}$ 2.2.c_c

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.a_i $3$ 2.64.a_ey 2.4.ae_m $4$ 2.256.bg_bdo 2.4.a_e $4$ 2.256.bg_bdo
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.a_i $3$ 2.64.a_ey 2.4.ae_m $4$ 2.256.bg_bdo 2.4.a_e $4$ 2.256.bg_bdo 2.4.e_m $4$ 2.256.bg_bdo 2.4.a_i $6$ (not in LMFDB) 2.4.ai_y $12$ (not in LMFDB) 2.4.ag_q $12$ (not in LMFDB) 2.4.ae_i $12$ (not in LMFDB) 2.4.ae_m $12$ (not in LMFDB) 2.4.ac_a $12$ (not in LMFDB) 2.4.ac_i $12$ (not in LMFDB) 2.4.a_ai $12$ (not in LMFDB) 2.4.a_e $12$ (not in LMFDB) 2.4.c_a $12$ (not in LMFDB) 2.4.c_i $12$ (not in LMFDB) 2.4.e_i $12$ (not in LMFDB) 2.4.e_m $12$ (not in LMFDB) 2.4.g_q $12$ (not in LMFDB) 2.4.i_y $12$ (not in LMFDB) 2.4.a_a $24$ (not in LMFDB) 2.4.ac_e $60$ (not in LMFDB) 2.4.c_e $60$ (not in LMFDB)