# Properties

 Label 4.2.ae_i_am_r Base Field $\F_{2}$ Dimension $4$ Ordinary Yes $p$-rank $4$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $4$ L-polynomial: $1 - 4 x + 8 x^{2} - 12 x^{3} + 17 x^{4} - 24 x^{5} + 32 x^{6} - 32 x^{7} + 16 x^{8}$ Frobenius angles: $\pm0.0755571399449$, $\pm0.203216343788$, $\pm0.424442860055$, $\pm0.703216343788$ Angle rank: $2$ (numerical) Number field: 8.0.18939904.2 Galois group: $D_4\times C_2$

This isogeny class is simple but not geometrically simple.

## Newton polygon

This isogeny class is ordinary.

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

## 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 $A(\F_{q^r})$ 2 292 2402 85264 930082 17183908 403078034 4278206464 59240758994 1097334925732

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 5 5 21 29 65 181 253 437 1025

## 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 $\F_{2^{4}}$ is 2.16.c_b 2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.1088.2$)$
All geometric endomorphisms are defined over $\F_{2^{4}}$.
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.a_c_a_b 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_i_m_r $2$ 4.4.a_c_a_b 4.2.ae_k_au_bh $8$ (not in LMFDB) 4.2.a_ac_a_b $8$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.2.e_i_m_r $2$ 4.4.a_c_a_b 4.2.ae_k_au_bh $8$ (not in LMFDB) 4.2.a_ac_a_b $8$ (not in LMFDB) 4.2.a_c_a_b $8$ (not in LMFDB) 4.2.e_k_u_bh $8$ (not in LMFDB) 4.2.ac_b_c_ad $24$ (not in LMFDB) 4.2.c_b_ac_ad $24$ (not in LMFDB)