# Properties

 Label 4.2.ae_g_ae_c Base Field $\F_{2}$ Dimension $4$ Ordinary No $p$-rank $0$ Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $4$ L-polynomial: $1 - 4 x + 6 x^{2} - 4 x^{3} + 2 x^{4} - 8 x^{5} + 24 x^{6} - 32 x^{7} + 16 x^{8}$ Frobenius angles: $\pm0.0377785699724$, $\pm0.148391828106$, $\pm0.398391828106$, $\pm0.787778569972$ Angle rank: $2$ (numerical) Number field: 8.0.18939904.2 Galois group: $D_4\times C_2$

This isogeny class is simple and geometrically simple.

## Newton polygon

 $p$-rank: $0$ Slopes: $[1/4, 1/4, 1/4, 1/4, 3/4, 3/4, 3/4, 3/4]$

## Point counts

This isogeny class does not contain a Jacobian, and it is unknown whether it is principally polarizable.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1 97 2689 67609 457601 19040809 323696297 4570976881 73309284673 955259018737

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 1 5 17 9 73 153 273 545 881

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is 8.0.18939904.2.
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{8}}$ is the simple isogeny class 4.256.q_ds_glk_jitk and its endomorphism algebra is the quaternion algebra over 4.0.1088.2 with the following ramification data at primes above $2$, and unramified at all archimedean places:
 $v$ ($2$,$$\pi$$) ($2$,$$\pi + 1$$) $\operatorname{inv}_v$ $1/2$ $1/2$
where $\pi$ is a root of $x^{4} - 2x^{3} + 5x^{2} - 4x + 2$.\n
All geometric endomorphisms are defined over $\F_{2^{8}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{2}}$  The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 4.4.ae_i_ai_e and its endomorphism algebra is 8.0.18939904.2.
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is the simple isogeny class 4.16.a_i_a_q and its endomorphism algebra is 8.0.18939904.2.

## 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 4.2.e_g_e_c $2$ 4.4.ae_i_ai_e 4.2.a_c_ae_c $4$ (not in LMFDB) 4.2.a_c_e_c $4$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.2.e_g_e_c $2$ 4.4.ae_i_ai_e 4.2.a_c_ae_c $4$ (not in LMFDB) 4.2.a_c_e_c $4$ (not in LMFDB) 4.2.a_c_ae_c $8$ (not in LMFDB) 4.2.a_c_e_c $8$ (not in LMFDB) 4.2.e_g_e_c $8$ (not in LMFDB) 4.2.a_ae_a_k $16$ (not in LMFDB) 4.2.a_e_a_k $16$ (not in LMFDB)