Properties

Label 3.5.ai_be_acy
Base field $\F_{5}$
Dimension $3$
$p$-rank $3$
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:  $3$
L-polynomial:  $( 1 - 4 x + 5 x^{2} )( 1 - 4 x + 9 x^{2} - 20 x^{3} + 25 x^{4} )$
  $1 - 8 x + 30 x^{2} - 76 x^{3} + 150 x^{4} - 200 x^{5} + 125 x^{6}$
Frobenius angles:  $\pm0.103885594917$, $\pm0.147583617650$, $\pm0.516810247272$
Angle rank:  $3$ (numerical)
Jacobians:  $0$
Isomorphism classes:  2

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $22$ $12980$ $1696288$ $226371200$ $31650524422$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $22$ $106$ $578$ $3238$ $16192$ $78846$ $391618$ $1958578$ $9777702$

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 $\times$ 2.5.ae_j 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
3.5.a_ac_ae$2$3.25.ae_aq_ko
3.5.a_ac_e$2$3.25.ae_aq_ko
3.5.i_be_cy$2$3.25.ae_aq_ko

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

TwistExtension degreeCommon base change
3.5.a_ac_ae$2$3.25.ae_aq_ko
3.5.a_ac_e$2$3.25.ae_aq_ko
3.5.i_be_cy$2$3.25.ae_aq_ko
3.5.ag_w_acg$4$(not in LMFDB)
3.5.ac_g_aw$4$(not in LMFDB)
3.5.c_g_w$4$(not in LMFDB)
3.5.g_w_cg$4$(not in LMFDB)