Abelian variety isogeny class 2.625.ado_eyw over $\F_{5^{4}}$

• Label: 2.625.ado_eyw
• Base field: $\F_{5^{4}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5^{4}}$
Dimension:	$2$
L-polynomial:	$( 1 - 25 x )^{2}( 1 - 42 x + 625 x^{2} )$
	$1 - 92 x + 3350 x^{2} - 57500 x^{3} + 390625 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.182554890994$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$336384$	$151900250112$	$59598153907020288$	$23283024052177838407680$	$9094946948231806682997318144$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$534$	$388862$	$244114038$	$152587626430$	$95367430911894$	$59604644753656382$	$37252902980255509878$	$23283064365119618051710$	$14551915228355909186373654$	$9094947017728928844467250302$

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{4}}$.

Endomorphism algebra over $\F_{5^{4}}$

The isogeny class factors as 1.625.aby $\times$ 1.625.abq and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.625.aby : the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.
• 1.625.abq : \(\Q(\sqrt{-46}) \).

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
2.625.ai_abgs	$2$	(not in LMFDB)
2.625.i_abgs	$2$	(not in LMFDB)
2.625.do_eyw	$2$	(not in LMFDB)
2.625.ar_hs	$3$	(not in LMFDB)