Properties

Label 4.5.al_ci_aii_vq
Base field $\F_{5}$
Dimension $4$
$p$-rank $4$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{5}$
Dimension:  $4$
L-polynomial:  $( 1 - 4 x + 5 x^{2} )( 1 - 7 x + 27 x^{2} - 73 x^{3} + 135 x^{4} - 175 x^{5} + 125 x^{6} )$
  $1 - 11 x + 60 x^{2} - 216 x^{3} + 562 x^{4} - 1080 x^{5} + 1500 x^{6} - 1375 x^{7} + 625 x^{8}$
Frobenius angles:  $\pm0.0229621162481$, $\pm0.147583617650$, $\pm0.333082169302$, $\pm0.478604549684$
Angle rank:  $4$ (numerical)

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $66$ $358380$ $245992626$ $139189057920$ $91763628038346$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-5$ $25$ $127$ $569$ $3005$ $15613$ $78164$ $390177$ $1953541$ $9769465$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5}$.

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

TwistExtension degreeCommon base change
4.5.ad_e_a_aw$2$(not in LMFDB)
4.5.d_e_a_aw$2$(not in LMFDB)
4.5.l_ci_ii_vq$2$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
4.5.ad_e_a_aw$2$(not in LMFDB)
4.5.d_e_a_aw$2$(not in LMFDB)
4.5.l_ci_ii_vq$2$(not in LMFDB)
4.5.aj_bu_agg_qa$4$(not in LMFDB)
4.5.af_s_acc_eu$4$(not in LMFDB)
4.5.f_s_cc_eu$4$(not in LMFDB)
4.5.j_bu_gg_qa$4$(not in LMFDB)