# Properties

 Label 4.5.al_cj_aio_wi Base Field $\F_{5}$ Dimension $4$ Ordinary No $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{5}$ Dimension: $4$ L-polynomial: $( 1 - 4 x + 5 x^{2} )( 1 - 7 x + 28 x^{2} - 75 x^{3} + 140 x^{4} - 175 x^{5} + 125 x^{6} )$ Frobenius angles: $\pm0.117658111351$, $\pm0.147583617650$, $\pm0.327130732663$, $\pm0.462990021908$ Angle rank: $4$ (numerical)

This isogeny class is not simple.

## Newton polygon

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

## 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 $A(\F_{q^r})$ 74 407740 280608296 151261754240 95397532510834 60862715950229440 37778120833792126274 23401661133629045399040 14585586808425220946415848 9106486304538120500625312700

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 27 142 619 3125 15954 79221 392611 1957642 9778007

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae $\times$ 3.5.ah_bc_acx and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{5}$.

## 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.ad_f_c_au $2$ (not in LMFDB) 4.5.d_f_ac_au $2$ (not in LMFDB) 4.5.l_cj_io_wi $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.ad_f_c_au $2$ (not in LMFDB) 4.5.d_f_ac_au $2$ (not in LMFDB) 4.5.l_cj_io_wi $2$ (not in LMFDB) 4.5.aj_bv_agk_qo $4$ (not in LMFDB) 4.5.af_t_acc_fa $4$ (not in LMFDB) 4.5.f_t_cc_fa $4$ (not in LMFDB) 4.5.j_bv_gk_qo $4$ (not in LMFDB)