Abelian variety isogeny class 2.37.at_gi over $\F_{37}$

• Label: 2.37.at_gi
• Base field: $\F_{37}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{37}$
Dimension:	$2$
L-polynomial:	$( 1 - 10 x + 37 x^{2} )( 1 - 9 x + 37 x^{2} )$
	$1 - 19 x + 164 x^{2} - 703 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.192861133077$, $\pm0.234922259125$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$812$	$1831872$	$2585105936$	$3521392890624$	$4810717835789852$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$19$	$1337$	$51034$	$1878913$	$69374719$	$2565844022$	$94931886691$	$3512475468001$	$129961703064658$	$4808584173938657$

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 $\F_{37}$.

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.ak $\times$ 1.37.aj and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.37.ak : \(\Q(\sqrt{-3}) \).
• 1.37.aj : \(\Q(\sqrt{-67}) \).

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
2.37.ab_aq	$2$	(not in LMFDB)
2.37.b_aq	$2$	(not in LMFDB)
2.37.t_gi	$2$	(not in LMFDB)
2.37.ak_df	$3$	(not in LMFDB)
2.37.c_az	$3$	(not in LMFDB)