# Properties

 Label 4.5.am_cu_akq_bcf 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: $( 1 - 3 x + 5 x^{2} )^{2}( 1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4} )$ Frobenius angles: $\pm0.0512862249088$, $\pm0.265942140215$, $\pm0.265942140215$, $\pm0.384619558242$ Angle rank: $2$ (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 $A(\F_{q^r})$ 63 403137 321076224 165538130625 93198017193183 57988092160180224 36809102946159389727 23197544263794809120625 14538268526711324681567232 9095618997158239136593946097

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 26 162 678 3054 15194 77190 389190 1951290 9766346

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ad 2 $\times$ 2.5.ag_r and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.5.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$ 2.5.ag_r : $$\Q(\sqrt{2}, \sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{6}}$ is 1.15625.afm 2 $\times$ 1.15625.acw 2 . The endomorphism algebra for each factor is: 1.15625.afm 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-6})$$$)$ 1.15625.acw 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$
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 1.25.b 2 $\times$ 2.25.ac_av. The endomorphism algebra for each factor is: 1.25.b 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$ 2.25.ac_av : $$\Q(\sqrt{2}, \sqrt{-3})$$.
• Endomorphism algebra over $\F_{5^{3}}$  The base change of $A$ to $\F_{5^{3}}$ is 1.125.s 2 $\times$ 2.125.a_afm. The endomorphism algebra for each factor is: 1.125.s 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$ 2.125.a_afm : $$\Q(\sqrt{2}, \sqrt{-3})$$.

## 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.ag_s_abk_cp $2$ (not in LMFDB) 4.5.a_a_am_n $2$ (not in LMFDB) 4.5.a_a_m_n $2$ (not in LMFDB) 4.5.g_s_bk_cp $2$ (not in LMFDB) 4.5.m_cu_kq_bcf $2$ (not in LMFDB) 4.5.ag_v_abq_dk $3$ (not in LMFDB) 4.5.ad_d_m_ack $3$ (not in LMFDB) 4.5.d_g_v_cg $3$ (not in LMFDB) 4.5.j_bn_eq_lm $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.ag_s_abk_cp $2$ (not in LMFDB) 4.5.a_a_am_n $2$ (not in LMFDB) 4.5.a_a_m_n $2$ (not in LMFDB) 4.5.g_s_bk_cp $2$ (not in LMFDB) 4.5.m_cu_kq_bcf $2$ (not in LMFDB) 4.5.ag_v_abq_dk $3$ (not in LMFDB) 4.5.ad_d_m_ack $3$ (not in LMFDB) 4.5.d_g_v_cg $3$ (not in LMFDB) 4.5.j_bn_eq_lm $3$ (not in LMFDB) 4.5.ag_q_ay_bh $4$ (not in LMFDB) 4.5.g_q_y_bh $4$ (not in LMFDB) 4.5.aj_bn_aeq_lm $6$ (not in LMFDB) 4.5.ad_g_av_cg $6$ (not in LMFDB) 4.5.a_d_a_ca $6$ (not in LMFDB) 4.5.d_d_am_ack $6$ (not in LMFDB) 4.5.g_v_bq_dk $6$ (not in LMFDB) 4.5.ag_r_as_m $12$ (not in LMFDB) 4.5.ad_c_aj_bq $12$ (not in LMFDB) 4.5.a_ad_a_ca $12$ (not in LMFDB) 4.5.a_ab_a_bw $12$ (not in LMFDB) 4.5.a_b_a_bw $12$ (not in LMFDB) 4.5.d_c_j_bq $12$ (not in LMFDB) 4.5.g_r_s_m $12$ (not in LMFDB) 4.5.ak_bz_ags_ra $24$ (not in LMFDB) 4.5.ah_y_acx_hu $24$ (not in LMFDB) 4.5.ae_h_aq_bq $24$ (not in LMFDB) 4.5.ae_j_ay_cg $24$ (not in LMFDB) 4.5.ac_d_s_abm $24$ (not in LMFDB) 4.5.ab_a_d_abm $24$ (not in LMFDB) 4.5.b_a_ad_abm $24$ (not in LMFDB) 4.5.c_d_as_abm $24$ (not in LMFDB) 4.5.e_h_q_bq $24$ (not in LMFDB) 4.5.e_j_y_cg $24$ (not in LMFDB) 4.5.h_y_cx_hu $24$ (not in LMFDB) 4.5.k_bz_gs_ra $24$ (not in LMFDB)