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

• Label: 2.37.at_gc
• 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: yes

Invariants

Base field:	$\F_{37}$
Dimension:	$2$
L-polynomial:	$( 1 - 12 x + 37 x^{2} )( 1 - 7 x + 37 x^{2} )$
	$1 - 19 x + 158 x^{2} - 703 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.0525684567113$, $\pm0.304847772502$
Angle rank:	$2$ (numerical)
Jacobians:	$3$
Isomorphism classes:	21

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$806$	$1813500$	$2567580704$	$3512386800000$	$4807713634644206$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$19$	$1325$	$50692$	$1874113$	$69331399$	$2565583850$	$94931066347$	$3512477811553$	$129961753668004$	$4808584520679125$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=27x^6+34x^5+3x^4+12x^3+11x^2+29x+27$
• $y^2=36x^6+12x^5+36x^4+6x^3+19x^2+15x+2$
• $y^2=29x^6+5x^5+34x^4+19x^3+31x^2+33x+24$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{37}$.

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.am $\times$ 1.37.ah 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.am : \(\Q(\sqrt{-1}) \).
• 1.37.ah : \(\Q(\sqrt{-11}) \).

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.af_ak	$2$	(not in LMFDB)
2.37.f_ak	$2$	(not in LMFDB)
2.37.t_gc	$2$	(not in LMFDB)