# Properties

 Label 5.2.ah_ba_acp_fd_aic Base Field $\F_{2}$ Dimension $5$ Ordinary No $p$-rank $4$ Principally polarizable Yes Contains a Jacobian No

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## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - 2 x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.123548644961$, $\pm0.174442860055$, $\pm0.250000000000$, $\pm0.456881978294$, $\pm0.546783656212$ Angle rank: $3$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $4$ Slopes: $[0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1]$

## 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 2660 61256 957600 69424562 2281173440 39982829098 1029278275200 35222880982952 1154645016587300

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 8 11 16 56 113 150 240 515 1048

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac $\times$ 2.2.ad_f $\times$ 2.2.ac_d 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 $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h 2 $\times$ 1.4096.ey $\times$ 2.4096.adu_hrl. The endomorphism algebra for each factor is: 1.4096.h 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 1.4096.ey : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.4096.adu_hrl : 4.0.1088.2.
All geometric endomorphisms are defined over $\F_{2^{12}}$.
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 $\times$ 2.4.b_ad $\times$ 2.4.c_b. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.e $\times$ 2.8.ac_p $\times$ 2.8.a_l. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 2.16.ah_bh $\times$ 2.16.ac_b. The endomorphism algebra for each factor is: 1.16.i : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.ah_bh : $$\Q(\sqrt{-3}, \sqrt{5})$$. 2.16.ac_b : 4.0.1088.2.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 1.64.l 2 $\times$ 2.64.ba_ld. The endomorphism algebra for each factor is: 1.64.a : $$\Q(\sqrt{-1})$$. 1.64.l 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 2.64.ba_ld : 4.0.1088.2.

## 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 5.2.ad_g_al_r_aw $2$ (not in LMFDB) 5.2.ad_g_ah_f_ac $2$ (not in LMFDB) 5.2.ab_c_ab_b_g $2$ (not in LMFDB) 5.2.b_c_b_b_ag $2$ (not in LMFDB) 5.2.d_g_h_f_c $2$ (not in LMFDB) 5.2.d_g_l_r_w $2$ (not in LMFDB) 5.2.h_ba_cp_fd_ic $2$ (not in LMFDB) 5.2.ae_i_ak_n_as $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ad_g_al_r_aw $2$ (not in LMFDB) 5.2.ad_g_ah_f_ac $2$ (not in LMFDB) 5.2.ab_c_ab_b_g $2$ (not in LMFDB) 5.2.b_c_b_b_ag $2$ (not in LMFDB) 5.2.d_g_h_f_c $2$ (not in LMFDB) 5.2.d_g_l_r_w $2$ (not in LMFDB) 5.2.h_ba_cp_fd_ic $2$ (not in LMFDB) 5.2.ae_i_ak_n_as $3$ (not in LMFDB) 5.2.a_a_ac_f_c $6$ (not in LMFDB) 5.2.a_a_c_f_ac $6$ (not in LMFDB) 5.2.e_i_k_n_s $6$ (not in LMFDB) 5.2.af_q_abn_cx_aem $8$ (not in LMFDB) 5.2.ab_e_ad_d_ae $8$ (not in LMFDB) 5.2.b_e_d_d_e $8$ (not in LMFDB) 5.2.f_q_bn_cx_em $8$ (not in LMFDB) 5.2.ae_k_as_bf_abu $12$ (not in LMFDB) 5.2.a_c_ac_h_ac $12$ (not in LMFDB) 5.2.a_c_c_h_c $12$ (not in LMFDB) 5.2.e_k_s_bf_bu $12$ (not in LMFDB) 5.2.ac_e_ag_j_ai $24$ (not in LMFDB) 5.2.ac_g_ak_t_ay $24$ (not in LMFDB) 5.2.c_e_g_j_i $24$ (not in LMFDB) 5.2.c_g_k_t_y $24$ (not in LMFDB)