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

• Label: 4.5.a_au_a_fu
• Base field: $\F_{5}$
• Dimension: $4$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$( 1 - 5 x^{2} )^{4}$
	$1 - 20 x^{2} + 150 x^{4} - 500 x^{6} + 625 x^{8}$
Frobenius angles:	$0$, $0$, $0$, $0$, $1$, $1$, $1$, $1$
Angle rank:	$0$ (numerical)
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$256$	$65536$	$236421376$	$110075314176$	$95245419909376$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$-14$	$126$	$426$	$3126$	$14626$	$78126$	$385626$	$1953126$	$9740626$

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_ak^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q(\sqrt{5}) \) ramified at both real infinite places.

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

The base change of $A$ to $\F_{5^{2}}$ is 1.25.ak^4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ is 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.a_af_a_a	$3$	(not in LMFDB)
4.5.a_k_a_cx	$3$	(not in LMFDB)
4.5.a_a_a_aby	$4$	(not in LMFDB)