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

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

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$( 1 - 5 x^{2} + 25 x^{4} )^{2}$
	$1 - 10 x^{2} + 75 x^{4} - 250 x^{6} + 625 x^{8}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.833333333333$, $\pm0.833333333333$
Angle rank:	$0$ (numerical)
Cyclic group of points:	no
Non-cyclic primes:	$3, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$441$	$194481$	$252047376$	$179607287601$	$95306425775001$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$6$	$126$	$726$	$3126$	$16626$	$78126$	$393126$	$1953126$	$9753126$

Jacobians and polarizations

This isogeny class is principally polarizable and contains no Jacobian of a hyperelliptic curve, but it is unknown whether it contains a Jacobian of a non-hyperelliptic curve.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 2.5.a_af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{-5})\)$)$

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

The base change of $A$ to $\F_{5^{6}}$ is 1.15625.jq^4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{5^{2}}$

  The base change of $A$ to $\F_{5^{2}}$ is 1.25.af^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$

• Endomorphism algebra over $\F_{5^{3}}$

  The base change of $A$ to $\F_{5^{3}}$ is 1.125.a^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-5}) \)$)$

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_f_a_a	$3$	(not in LMFDB)
4.5.a_u_a_fu	$3$	(not in LMFDB)
4.5.a_a_a_z	$4$	(not in LMFDB)