# Properties

 Label 2.4.a_a 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: $1 + 16 x^{4}$ Frobenius angles: $\pm0.250000000000$, $\pm0.750000000000$ Angle rank: $0$ (numerical) Number field: $$\Q(\zeta_{8})$$ Galois group: $C_2^2$ Jacobians: 2

This isogeny class is simple but not geometrically 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 17 289 4097 83521 1048577 16785409 268435457 4228250625 68719476737 1099513724929

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 17 65 321 1025 4097 16385 64513 262145 1048577

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\zeta_{8})$$.
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{8}}$ is 1.256.bg 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.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$

## 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.ai_y $8$ (not in LMFDB) 2.4.ae_i $8$ (not in LMFDB) 2.4.a_ai $8$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.ai_y $8$ (not in LMFDB) 2.4.ae_i $8$ (not in LMFDB) 2.4.a_ai $8$ (not in LMFDB) 2.4.a_i $8$ (not in LMFDB) 2.4.e_i $8$ (not in LMFDB) 2.4.i_y $8$ (not in LMFDB) 2.4.ag_q $24$ (not in LMFDB) 2.4.ae_m $24$ (not in LMFDB) 2.4.ac_a $24$ (not in LMFDB) 2.4.ac_i $24$ (not in LMFDB) 2.4.a_ae $24$ (not in LMFDB) 2.4.a_e $24$ (not in LMFDB) 2.4.c_a $24$ (not in LMFDB) 2.4.c_i $24$ (not in LMFDB) 2.4.e_m $24$ (not in LMFDB) 2.4.g_q $24$ (not in LMFDB) 2.4.ac_e $40$ (not in LMFDB) 2.4.c_e $40$ (not in LMFDB)