Abelian variety isogeny class 4.5.al_ci_aif_ve over $\F_{5}$

• Label: 4.5.al_ci_aif_ve
• Base field: $\F_{5}$
• Dimension: $4$
• $p$-rank: $3$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$( 1 - 3 x + 5 x^{2} )( 1 + 5 x^{2} )( 1 - 4 x + 5 x^{2} )^{2}$
	$1 - 11 x + 60 x^{2} - 213 x^{3} + 550 x^{4} - 1065 x^{5} + 1500 x^{6} - 1375 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.147583617650$, $\pm0.147583617650$, $\pm0.265942140215$, $\pm0.5$
Angle rank:	$2$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$72$	$388800$	$270055296$	$159252480000$	$102015994462632$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$25$	$136$	$653$	$3335$	$16270$	$78731$	$390333$	$1955080$	$9775825$

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 $\F_{5^{2}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ae^2 $\times$ 1.5.ad $\times$ 1.5.a 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.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 1.5.ad : \(\Q(\sqrt{-11}) \).
• 1.5.a : \(\Q(\sqrt{-5}) \).

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

The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag^2 $\times$ 1.25.b $\times$ 1.25.k. The endomorphism algebra for each factor is:

• 1.25.ag^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 1.25.b : \(\Q(\sqrt{-11}) \).
• 1.25.k : the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.

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.af_m_abb_cs	$2$	(not in LMFDB)
4.5.ad_e_d_ak	$2$	(not in LMFDB)
4.5.d_e_ad_ak	$2$	(not in LMFDB)
4.5.f_m_bb_cs	$2$	(not in LMFDB)
4.5.l_ci_if_ve	$2$	(not in LMFDB)
4.5.b_j_m_bo	$3$	(not in LMFDB)