# Properties

 Label 2.4.ae_i 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: $( 1 - 2 x )^{2}( 1 + 4 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.5$ 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:

• $y^2+y=x^5+x^4+a$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 5 225 3185 50625 985025 16769025 264273665 4228250625 68451564545 1099509530625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 1 17 49 193 961 4097 16129 64513 261121 1048577

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The isogeny class factors as 1.4.ae $\times$ 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: 1.4.ae : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.4.a : $$\Q(\sqrt{-1})$$.
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg 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 $\F_{2^{8}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.i. The endomorphism algebra for each factor is: 1.16.ai : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.16.i : 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.e_i $2$ 2.16.a_abg 2.4.c_i $3$ 2.64.aq_ey 2.4.ai_y $4$ 2.256.acm_chc
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.e_i $2$ 2.16.a_abg 2.4.c_i $3$ 2.64.aq_ey 2.4.ai_y $4$ 2.256.acm_chc 2.4.a_ai $4$ 2.256.acm_chc 2.4.a_i $4$ 2.256.acm_chc 2.4.i_y $4$ 2.256.acm_chc 2.4.ac_i $6$ (not in LMFDB) 2.4.a_a $8$ (not in LMFDB) 2.4.ag_q $12$ (not in LMFDB) 2.4.ae_m $12$ (not in LMFDB) 2.4.ac_a $12$ (not in LMFDB) 2.4.a_ae $12$ (not in LMFDB) 2.4.a_e $12$ (not in LMFDB) 2.4.c_a $12$ (not in LMFDB) 2.4.e_m $12$ (not in LMFDB) 2.4.g_q $12$ (not in LMFDB) 2.4.ac_e $20$ (not in LMFDB) 2.4.c_e $20$ (not in LMFDB)