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

• Label: 3.5.ai_bi_adq
• 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

Invariants

Base field:	$\F_{5}$
Dimension:	$3$
L-polynomial:	$( 1 - 2 x + 5 x^{2} )( 1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4} )$
	$1 - 8 x + 34 x^{2} - 94 x^{3} + 170 x^{4} - 200 x^{5} + 125 x^{6}$
Frobenius angles:	$\pm0.0512862249088$, $\pm0.352416382350$, $\pm0.384619558242$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$28$	$17696$	$2291632$	$232525440$	$28001290268$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$30$	$148$	$594$	$2858$	$15108$	$78146$	$392546$	$1955524$	$9759870$

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 $\F_{5^{6}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ac $\times$ 2.5.ag_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.ac : \(\Q(\sqrt{-1}) \).
• 2.5.ag_r : \(\Q(\sqrt{2}, \sqrt{-3})\).

Endomorphism algebra over $\overline{\F}_{5}$

The base change of $A$ to $\F_{5^{6}}$ is 1.15625.aja $\times$ 1.15625.afm^2. The endomorphism algebra for each factor is:

• 1.15625.aja : \(\Q(\sqrt{-1}) \).
• 1.15625.afm^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{5^{2}}$
  The base change of $A$ to $\F_{5^{2}}$ is 1.25.g $\times$ 2.25.ac_av. The endomorphism algebra for each factor is:

  • 1.25.g : \(\Q(\sqrt{-1}) \).
  • 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.w $\times$ 2.125.a_afm. The endomorphism algebra for each factor is:

  • 1.125.w : \(\Q(\sqrt{-1}) \).
  • 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.ae_k_aba	$2$	3.25.e_ai_ais
3.5.e_k_ba	$2$	3.25.e_ai_ais
3.5.i_bi_dq	$2$	3.25.e_ai_ais
3.5.ac_h_ae	$3$	(not in LMFDB)
3.5.e_k_ba	$3$	(not in LMFDB)