# Properties

 Label 4.5.al_ci_aik_vz 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: $( 1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4} )( 1 - 5 x + 13 x^{2} - 25 x^{3} + 25 x^{4} )$ Frobenius angles: $\pm0.0512862249088$, $\pm0.0878807261908$, $\pm0.384619558242$, $\pm0.450170915301$ Angle rank: $3$ (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 343413 233700012 128379112029 87029776331568 59175145450918224 37623882916287319491 23355899060757619121325 14547860733152098732413852 9095463641330437680639848448

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 25 121 517 2840 15511 78899 391845 1952581 9766180

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 2.5.ag_r $\times$ 2.5.af_n 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 $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{6}}$ is 1.15625.afm 2 $\times$ 2.15625.gn_byxl. The endomorphism algebra for each factor is: 1.15625.afm 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-6})$$$)$ 2.15625.gn_byxl : 4.0.4901.1.
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 $\times$ 2.25.b_abf. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{5^{3}}$  The base change of $A$ to $\F_{5^{3}}$ is 2.125.af_dt $\times$ 2.125.a_afm. The endomorphism algebra for each factor is:

## 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.ab_a_c_abd $2$ (not in LMFDB) 4.5.b_a_ac_abd $2$ (not in LMFDB) 4.5.l_ci_ik_vz $2$ (not in LMFDB) 4.5.af_p_abj_cy $3$ (not in LMFDB) 4.5.b_a_ac_abd $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.ab_a_c_abd $2$ (not in LMFDB) 4.5.b_a_ac_abd $2$ (not in LMFDB) 4.5.l_ci_ik_vz $2$ (not in LMFDB) 4.5.af_p_abj_cy $3$ (not in LMFDB) 4.5.b_a_ac_abd $3$ (not in LMFDB) 4.5.f_p_bj_cy $6$ (not in LMFDB) 4.5.af_l_ap_y $12$ (not in LMFDB) 4.5.f_l_p_y $12$ (not in LMFDB) 4.5.aj_bp_afh_nq $24$ (not in LMFDB) 4.5.ab_b_h_abu $24$ (not in LMFDB) 4.5.b_b_ah_abu $24$ (not in LMFDB) 4.5.j_bp_fh_nq $24$ (not in LMFDB)