# Properties

 Label 4.5.am_cs_ake_bax Base Field $\F_{5}$ Dimension $4$ Ordinary Yes $p$-rank $4$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{5}$ Dimension: $4$ L-polynomial: $( 1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4} )^{2}$ Frobenius angles: $\pm0.0512862249088$, $\pm0.0512862249088$, $\pm0.384619558242$, $\pm0.384619558242$ Angle rank: $1$ (numerical)

This isogeny class is not 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 $A(\F_{q^r})$ 49 305809 239754256 132002149041 84618786129409 57482103270113536 37328307093883814449 23386524211874297677329 14551944166392407808553744 9085114642646574814495668049

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 22 126 534 2754 15058 78282 392358 1953126 9755062

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 2.5.ag_r 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{2}, \sqrt{-3})$$$)$
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{6}}$ is 1.15625.afm 4 and its endomorphism algebra is $\mathrm{M}_{4}($$$\Q(\sqrt{-6})$$$)$
All geometric endomorphisms are defined over $\F_{5^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{5^{2}}$  The base change of $A$ to $\F_{5^{2}}$ is 2.25.ac_av 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{2}, \sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{5^{3}}$  The base change of $A$ to $\F_{5^{3}}$ is 2.125.a_afm 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{2}, \sqrt{-3})$$$)$

## 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 4.5.a_ac_a_av $2$ (not in LMFDB) 4.5.m_cs_ke_bax $2$ (not in LMFDB) 4.5.ag_t_abq_dg $3$ (not in LMFDB) 4.5.a_ac_a_av $3$ (not in LMFDB) 4.5.a_e_a_cc $3$ (not in LMFDB) 4.5.g_t_bq_dg $3$ (not in LMFDB) 4.5.m_cs_ke_bax $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.a_ac_a_av $2$ (not in LMFDB) 4.5.m_cs_ke_bax $2$ (not in LMFDB) 4.5.ag_t_abq_dg $3$ (not in LMFDB) 4.5.a_ac_a_av $3$ (not in LMFDB) 4.5.a_e_a_cc $3$ (not in LMFDB) 4.5.g_t_bq_dg $3$ (not in LMFDB) 4.5.m_cs_ke_bax $3$ (not in LMFDB) 4.5.a_c_a_av $4$ (not in LMFDB) 4.5.ae_i_i_abp $8$ (not in LMFDB) 4.5.e_i_ai_abp $8$ (not in LMFDB) 4.5.ag_p_as_q $12$ (not in LMFDB) 4.5.a_ae_a_cc $12$ (not in LMFDB) 4.5.a_a_a_bu $12$ (not in LMFDB) 4.5.g_p_s_q $12$ (not in LMFDB) 4.5.ak_bx_agk_qk $24$ (not in LMFDB) 4.5.ai_bg_aea_ko $24$ (not in LMFDB) 4.5.ae_g_am_bi $24$ (not in LMFDB) 4.5.ae_k_abc_co $24$ (not in LMFDB) 4.5.ac_b_k_acc $24$ (not in LMFDB) 4.5.a_a_a_abu $24$ (not in LMFDB) 4.5.c_b_ak_acc $24$ (not in LMFDB) 4.5.e_g_m_bi $24$ (not in LMFDB) 4.5.e_k_bc_co $24$ (not in LMFDB) 4.5.i_bg_ea_ko $24$ (not in LMFDB) 4.5.k_bx_gk_qk $24$ (not in LMFDB)