Abelian variety isogeny class 3.5.j_bo_eh over $\F_{5}$

• Label: 3.5.j_bo_eh
• 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: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$3$
L-polynomial:	$( 1 + 3 x + 5 x^{2} )( 1 + 6 x + 17 x^{2} + 30 x^{3} + 25 x^{4} )$
	$1 + 9 x + 40 x^{2} + 111 x^{3} + 200 x^{4} + 225 x^{5} + 125 x^{6}$
Frobenius angles:	$\pm0.615380441758$, $\pm0.734057859785$, $\pm0.948713775091$
Angle rank:	$2$ (numerical)
Jacobians:	$1$
Cyclic group of points:	yes

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

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$711$	$14931$	$1672272$	$245241675$	$31800115611$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$15$	$25$	$108$	$629$	$3255$	$15268$	$78555$	$390341$	$1954044$	$9763345$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is not hyperelliptic):

• $3 x^3 y+3 x^3 z+3 x^2 y^2+2 x^2 y z+x^2 z^2+x z^3+y^3 z=0$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{6}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.d $\times$ 2.5.g_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.d : \(\Q(\sqrt{-11}) \).
• 2.5.g_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. The endomorphism algebra for each factor is:

• 1.15625.afm^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$
• 1.15625.acw : \(\Q(\sqrt{-11}) \).

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 $\times$ 2.25.ac_av. The endomorphism algebra for each factor is:

  • 1.25.b : \(\Q(\sqrt{-11}) \).
  • 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.as $\times$ 2.125.a_afm. The endomorphism algebra for each factor is:

  • 1.125.as : \(\Q(\sqrt{-11}) \).
  • 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
3.5.aj_bo_aeh	$2$	3.25.ab_c_aer
3.5.ad_e_aj	$2$	3.25.ab_c_aer
3.5.d_e_j	$2$	3.25.ab_c_aer
3.5.ad_e_aj	$3$	(not in LMFDB)
3.5.d_h_g	$3$	(not in LMFDB)