# Properties

 Label 2.64.ay_jw Base Field $\F_{2^{6}}$ Dimension $2$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2^{6}}$ Dimension: $2$ L-polynomial: $( 1 - 8 x )^{2}( 1 - 8 x + 64 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.333333333333$ 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 $A(\F_{q^r})$ 2793 16515009 68718952449 281406257229825 1152815955785449473 4722294425687923097601 19342785443735548410724353 79228157791897854723881959425 324518553658426690754359001612289 1329227994546975833618426784307478529

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 41 4033 262145 16773121 1073643521 68718428161 4398040219649 281474959933441 18014398509481985 1152921503533105153

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
 The isogeny class factors as 1.64.aq $\times$ 1.64.ai and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.64.aq : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.64.ai : $$\Q(\sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{2^{6}}$
 The base change of $A$ to $\F_{2^{36}}$ is 1.68719476736.abdvoy 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 $\F_{2^{36}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{12}}$  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey $\times$ 1.4096.cm. The endomorphism algebra for each factor is: 1.4096.aey : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.4096.cm : $$\Q(\sqrt{-3})$$.
• Endomorphism algebra over $\F_{2^{18}}$  The base change of $A$ to $\F_{2^{18}}$ is 1.262144.abnk $\times$ 1.262144.bnk. The endomorphism algebra for each factor is: 1.262144.abnk : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.262144.bnk : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.

## 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 2.64.ai_a $2$ (not in LMFDB) 2.64.i_a $2$ (not in LMFDB) 2.64.y_jw $2$ (not in LMFDB) 2.64.a_aey $3$ (not in LMFDB) 2.64.a_cm $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.64.ai_a $2$ (not in LMFDB) 2.64.i_a $2$ (not in LMFDB) 2.64.y_jw $2$ (not in LMFDB) 2.64.a_aey $3$ (not in LMFDB) 2.64.a_cm $3$ (not in LMFDB) 2.64.ai_ey $4$ (not in LMFDB) 2.64.i_ey $4$ (not in LMFDB) 2.64.abg_ou $6$ (not in LMFDB) 2.64.aq_hk $6$ (not in LMFDB) 2.64.ai_a $6$ (not in LMFDB) 2.64.q_hk $6$ (not in LMFDB) 2.64.bg_ou $6$ (not in LMFDB) 2.64.aq_ey $12$ (not in LMFDB) 2.64.a_acm $12$ (not in LMFDB) 2.64.a_ey $12$ (not in LMFDB) 2.64.q_ey $12$ (not in LMFDB) 2.64.a_a $24$ (not in LMFDB) 2.64.ai_cm $30$ (not in LMFDB) 2.64.i_cm $30$ (not in LMFDB)