# Properties

 Label 2.4.ai_y 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: $( 1 - 2 x )^{4}$ Frobenius angles: $0$, $0$, $0$, $0$ 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 $A(\F_{q^r})$ 1 81 2401 50625 923521 15752961 260144641 4228250625 68184176641 1095222947841

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 1 33 193 897 3841 15873 64513 260097 1044481

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The isogeny class factors as 1.4.ae 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 $\F_{2^{2}}$.

## 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_ae

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.a_ai $2$ 2.16.aq_ds 2.4.i_y $2$ 2.16.aq_ds 2.4.ac_a $3$ 2.64.abg_ou 2.4.e_m $3$ 2.64.abg_ou
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.a_ai $2$ 2.16.aq_ds 2.4.i_y $2$ 2.16.aq_ds 2.4.ac_a $3$ 2.64.abg_ou 2.4.e_m $3$ 2.64.abg_ou 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.c_e $5$ 2.1024.aey_jci 2.4.ag_q $6$ (not in LMFDB) 2.4.ae_m $6$ (not in LMFDB) 2.4.a_e $6$ (not in LMFDB) 2.4.c_a $6$ (not in LMFDB) 2.4.g_q $6$ (not in LMFDB) 2.4.a_a $8$ (not in LMFDB) 2.4.ac_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)