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

• Label: 4.5.a_af_a_a
• 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:	$( 1 - 5 x^{2} )^{2}( 1 + 5 x^{2} + 25 x^{4} )$
	$1 - 5 x^{2} - 125 x^{6} + 625 x^{8}$
Frobenius angles:	$0$, $0$, $\pm0.333333333333$, $\pm0.666666666667$, $1$, $1$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$496$	$246016$	$236421376$	$140607000576$	$95336914059376$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$16$	$126$	$576$	$3126$	$14626$	$78126$	$389376$	$1953126$	$9759376$

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_ak $\times$ 2.5.a_f and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 2.5.a_ak : the quaternion algebra over \(\Q(\sqrt{5}) \) ramified at both real infinite places.
• 2.5.a_f : \(\Q(\sqrt{-3}, \sqrt{5})\).

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

The base change of $A$ to $\F_{5^{6}}$ is 1.15625.ajq^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.ak^2 $\times$ 1.25.f^2. The endomorphism algebra for each factor is:

  • 1.25.ak^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.
  • 1.25.f^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• Endomorphism algebra over $\F_{5^{3}}$

  The base change of $A$ to $\F_{5^{3}}$ is 2.125.a_ajq^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.

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_au_a_fu	$3$	(not in LMFDB)
4.5.a_k_a_cx	$3$	(not in LMFDB)
4.5.a_ap_a_dw	$4$	(not in LMFDB)