# Properties

 Label 5.2.ag_s_abi_bw_acm Base Field $\F_{2}$ Dimension $5$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x + 2 x^{2} - 2 x^{3} + 4 x^{4} - 8 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.111901318694$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.359194778829$, $\pm0.729359314356$ Angle rank: $3$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $0$ Slopes: $[1/3, 1/3, 1/3, 1/2, 1/2, 1/2, 1/2, 2/3, 2/3, 2/3]$

## 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 $A(\F_{q^r})$ 3 2025 65403 4505625 39491733 927087525 36600767913 955088870625 36862172735193 1220220502700625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 5 15 41 37 53 137 225 537 1105

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 3.2.ac_c_ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.2.ac 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 3.2.ac_c_ac : 6.0.1142512.1.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 3.16.i_bw_jg. The endomorphism algebra for each factor is: 1.16.i 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 3.16.i_bw_jg : 6.0.1142512.1.
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 1.4.a 2 $\times$ 3.4.a_e_ae. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 3.4.a_e_ae : 6.0.1142512.1.

## 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 5.2.ac_c_ac_i_aq $2$ (not in LMFDB) 5.2.ac_c_c_a_a $2$ (not in LMFDB) 5.2.c_c_ac_a_a $2$ (not in LMFDB) 5.2.c_c_c_i_q $2$ (not in LMFDB) 5.2.g_s_bi_bw_cm $2$ (not in LMFDB) 5.2.a_a_c_a_ae $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ac_c_ac_i_aq $2$ (not in LMFDB) 5.2.ac_c_c_a_a $2$ (not in LMFDB) 5.2.c_c_ac_a_a $2$ (not in LMFDB) 5.2.c_c_c_i_q $2$ (not in LMFDB) 5.2.g_s_bi_bw_cm $2$ (not in LMFDB) 5.2.a_a_c_a_ae $3$ (not in LMFDB) 5.2.ae_i_ao_y_abk $6$ (not in LMFDB) 5.2.a_a_ac_a_e $6$ (not in LMFDB) 5.2.e_i_o_y_bk $6$ (not in LMFDB) 5.2.ae_k_as_bc_abo $8$ (not in LMFDB) 5.2.ac_ac_g_a_ai $8$ (not in LMFDB) 5.2.ac_g_ak_q_ay $8$ (not in LMFDB) 5.2.a_c_ac_e_ai $8$ (not in LMFDB) 5.2.a_c_c_e_i $8$ (not in LMFDB) 5.2.c_ac_ag_a_i $8$ (not in LMFDB) 5.2.c_g_k_q_y $8$ (not in LMFDB) 5.2.e_k_s_bc_bo $8$ (not in LMFDB) 5.2.ac_a_c_e_am $24$ (not in LMFDB) 5.2.ac_e_ag_m_au $24$ (not in LMFDB) 5.2.c_a_ac_e_m $24$ (not in LMFDB) 5.2.c_e_g_m_u $24$ (not in LMFDB)