# Properties

 Label 2.4.ac_i 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: $( 1 - 2 x + 4 x^{2} )( 1 + 4 x^{2} )$ Frobenius angles: $\pm0.333333333333$, $\pm0.5$ 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 $A(\F_{q^r})$ 15 525 5265 61425 1017825 16769025 266370945 4278189825 68988437505 1102737050625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 29 81 241 993 4097 16257 65281 263169 1051649

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The isogeny class factors as 1.4.ac $\times$ 1.4.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.amdc 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 $\F_{2^{24}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.e $\times$ 1.16.i. The endomorphism algebra for each factor is: 1.16.e : $$\Q(\sqrt{-3})$$. 1.16.i : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 1.64.q. The endomorphism algebra for each factor is: 1.64.a : $$\Q(\sqrt{-1})$$. 1.64.q : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
• Endomorphism algebra over $\F_{2^{8}}$  The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg $\times$ 1.256.q. The endomorphism algebra for each factor is: 1.256.abg : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.256.q : $$\Q(\sqrt{-3})$$.
• Endomorphism algebra over $\F_{2^{12}}$  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey $\times$ 1.4096.ey. The endomorphism algebra for each factor is: 1.4096.aey : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.4096.ey : 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 2.4.c_i $2$ 2.16.m_cm 2.4.e_i $3$ 2.64.q_ey 2.4.ag_q $4$ 2.256.aq_a
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.c_i $2$ 2.16.m_cm 2.4.e_i $3$ 2.64.q_ey 2.4.ag_q $4$ 2.256.aq_a 2.4.ac_a $4$ 2.256.aq_a 2.4.c_a $4$ 2.256.aq_a 2.4.g_q $4$ 2.256.aq_a 2.4.ae_i $6$ (not in LMFDB) 2.4.ai_y $12$ (not in LMFDB) 2.4.ae_m $12$ (not in LMFDB) 2.4.a_ai $12$ (not in LMFDB) 2.4.a_ae $12$ (not in LMFDB) 2.4.a_e $12$ (not in LMFDB) 2.4.a_i $12$ (not in LMFDB) 2.4.e_m $12$ (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)