Abelian variety isogeny class 2.25.ap_dw over $\F_{5^{2}}$

• Label: 2.25.ap_dw
• Base field: $\F_{5^{2}}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{5^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x )^{2}( 1 - 5 x + 25 x^{2} )$
	$1 - 15 x + 100 x^{2} - 375 x^{3} + 625 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.333333333333$
Angle rank:	$0$ (numerical)
Jacobians:	$0$

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:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$336$	$374976$	$244109376$	$152343749376$	$95275917959376$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$601$	$15626$	$390001$	$9756251$	$244078126$	$6103281251$	$152587500001$	$3814697265626$	$95367421875001$

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 $\F_{5^{12}}$.

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

The isogeny class factors as 1.25.ak $\times$ 1.25.af and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.25.ak : the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.
• 1.25.af : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{5^{2}}$

The base change of $A$ to $\F_{5^{12}}$ is 1.244140625.abufy^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{5^{4}}$
  The base change of $A$ to $\F_{5^{4}}$ is 1.625.aby $\times$ 1.625.z. The endomorphism algebra for each factor is:

  • 1.625.aby : the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.
  • 1.625.z : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{5^{6}}$
  The base change of $A$ to $\F_{5^{6}}$ is 1.15625.ajq $\times$ 1.15625.jq. The endomorphism algebra for each factor is:

  • 1.15625.ajq : the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.
  • 1.15625.jq : 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
2.25.af_a	$2$	2.625.az_a
2.25.f_a	$2$	2.625.az_a
2.25.p_dw	$2$	2.625.az_a
2.25.a_aby	$3$	(not in LMFDB)
2.25.a_z	$3$	(not in LMFDB)
2.25.p_dw	$3$	(not in LMFDB)