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

• Label: 4.5.a_aq_a_ek
• 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

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$( 1 - 8 x^{2} + 25 x^{4} )^{2}$
	$1 - 16 x^{2} + 114 x^{4} - 400 x^{6} + 625 x^{8}$
Frobenius angles:	$\pm0.102416382350$, $\pm0.102416382350$, $\pm0.897583617650$, $\pm0.897583617650$
Angle rank:	$1$ (numerical)
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$324$	$104976$	$246929796$	$140283207936$	$95489208772164$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$-6$	$126$	$570$	$3126$	$15978$	$78126$	$394842$	$1953126$	$9790554$

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 2.5.a_ai^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\zeta_{8})\)$)$

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

The base change of $A$ to $\F_{5^{2}}$ is 1.25.ai^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$

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.a_i_a_bn	$3$	(not in LMFDB)
4.5.a_a_a_ao	$4$	(not in LMFDB)
4.5.a_q_a_ek	$4$	(not in LMFDB)