# Properties

 Label 2.4.ae_m 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: $( 1 - 2 x + 4 x^{2} )^{2}$ Frobenius angles: $\pm0.333333333333$, $\pm0.333333333333$ Angle rank: $0$ (numerical) Jacobians: 1

This isogeny class is not 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:

• $y^2+y=x^5+x^4+x^3+a+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 9 441 6561 74529 986049 15752961 264290049 4328718849 69257922561 1101662259201

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 1 25 97 289 961 3841 16129 66049 264193 1050625

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The isogeny class factors as 1.4.ac 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{6}}$ is 1.64.q 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 $\F_{2^{6}}$.

## 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.a_ac

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.a_e $2$ 2.16.i_bw 2.4.e_m $2$ 2.16.i_bw 2.4.c_a $3$ 2.64.bg_ou 2.4.i_y $3$ 2.64.bg_ou
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.a_e $2$ 2.16.i_bw 2.4.e_m $2$ 2.16.i_bw 2.4.c_a $3$ 2.64.bg_ou 2.4.i_y $3$ 2.64.bg_ou 2.4.a_ae $4$ 2.256.bg_bdo 2.4.ai_y $6$ (not in LMFDB) 2.4.ag_q $6$ (not in LMFDB) 2.4.ac_a $6$ (not in LMFDB) 2.4.a_ai $6$ (not in LMFDB) 2.4.g_q $6$ (not in LMFDB) 2.4.ae_i $12$ (not in LMFDB) 2.4.ac_i $12$ (not in LMFDB) 2.4.a_ae $12$ (not in LMFDB) 2.4.a_i $12$ (not in LMFDB) 2.4.c_i $12$ (not in LMFDB) 2.4.e_i $12$ (not in LMFDB) 2.4.ac_e $15$ (not in LMFDB) 2.4.a_a $24$ (not in LMFDB) 2.4.c_e $30$ (not in LMFDB)