# Properties

 Label 2.4.a_e 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: $( 1 - 2 x + 4 x^{2} )( 1 + 2 x + 4 x^{2} )$ Frobenius angles: $\pm0.333333333333$, $\pm0.666666666667$ Angle rank: $0$ (numerical) Jacobians: 2

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 2 curves, and hence is principally polarizable:

• $y^2+y=x^5+ax^4+x^3$
• $y^2+y=x^5+(a+1)x^4+x^3$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21 441 3969 74529 1049601 15752961 268451841 4328718849 68718952449 1101662259201

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 25 65 289 1025 3841 16385 66049 262145 1050625

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The isogeny class factors as 1.4.ac $\times$ 1.4.c 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 $\F_{2^{12}}$ is 1.4096.aey 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 $\F_{2^{12}}$.
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 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aq $\times$ 1.64.q. The endomorphism algebra for each factor is: 1.64.aq : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.64.q : 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.ae_m $2$ 2.16.i_bw 2.4.e_m $2$ 2.16.i_bw 2.4.ag_q $3$ 2.64.a_aey 2.4.a_ai $3$ 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.ae_m $2$ 2.16.i_bw 2.4.e_m $2$ 2.16.i_bw 2.4.ag_q $3$ 2.64.a_aey 2.4.a_ai $3$ 2.64.a_aey 2.4.g_q $3$ 2.64.a_aey 2.4.a_ae $4$ 2.256.bg_bdo 2.4.ai_y $6$ (not in LMFDB) 2.4.ac_a $6$ (not in LMFDB) 2.4.c_a $6$ (not in LMFDB) 2.4.i_y $6$ (not in LMFDB) 2.4.ae_i $12$ (not in LMFDB) 2.4.ac_i $12$ (not in LMFDB) 2.4.a_ae $12$ (not in LMFDB) 2.4.a_i $12$ (not in LMFDB) 2.4.c_i $12$ (not in LMFDB) 2.4.e_i $12$ (not in LMFDB) 2.4.a_a $24$ (not in LMFDB) 2.4.ac_e $30$ (not in LMFDB) 2.4.c_e $30$ (not in LMFDB)