# Properties

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

## Invariants

 Base field: $\F_{2^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 2 x )^{2}( 1 + 2 x )^{2}$ Frobenius angles: $0$, $0$, $1$, $1$ Angle rank: $0$ (numerical) Jacobians: 0

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 is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 9 81 3969 50625 1046529 15752961 268402689 4228250625 68718952449 1095222947841

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 1 65 193 1025 3841 16385 64513 262145 1044481

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The isogeny class factors as 1.4.ae $\times$ 1.4.e and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.4.ae : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.4.e : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai 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 $\F_{2^{4}}$.

## 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 2.4.ai_y $2$ 2.16.aq_ds 2.4.i_y $2$ 2.16.aq_ds 2.4.ag_q $3$ 2.64.a_aey 2.4.a_e $3$ 2.64.a_aey 2.4.g_q $3$ 2.64.a_aey
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.ai_y $2$ 2.16.aq_ds 2.4.i_y $2$ 2.16.aq_ds 2.4.ag_q $3$ 2.64.a_aey 2.4.a_e $3$ 2.64.a_aey 2.4.g_q $3$ 2.64.a_aey 2.4.ae_i $4$ 2.256.acm_chc 2.4.a_i $4$ 2.256.acm_chc 2.4.e_i $4$ 2.256.acm_chc 2.4.ae_m $6$ (not in LMFDB) 2.4.ac_a $6$ (not in LMFDB) 2.4.c_a $6$ (not in LMFDB) 2.4.e_m $6$ (not in LMFDB) 2.4.a_a $8$ (not in LMFDB) 2.4.ac_e $10$ (not in LMFDB) 2.4.c_e $10$ (not in LMFDB) 2.4.ac_i $12$ (not in LMFDB) 2.4.a_ae $12$ (not in LMFDB) 2.4.c_i $12$ (not in LMFDB)