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

• Label: 2.37.av_ha
• 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:	$( 1 - 12 x + 37 x^{2} )( 1 - 9 x + 37 x^{2} )$
	$1 - 21 x + 182 x^{2} - 777 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.0525684567113$, $\pm0.234922259125$
Angle rank:	$2$ (numerical)
Jacobians:	$1$
Isomorphism classes:	3

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$754$	$1771900$	$2559338392$	$3513465072000$	$4808786765648554$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$17$	$1293$	$50528$	$1874689$	$69346877$	$2565699306$	$94931358665$	$3512475045601$	$129961716525776$	$4808584320331893$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=17x^6+5x^5+34x^4+2x^3+4x^2+x+32$

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.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.am : \(\Q(\sqrt{-1}) \).
• 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.ad_abi	$2$	(not in LMFDB)
2.37.d_abi	$2$	(not in LMFDB)
2.37.v_ha	$2$	(not in LMFDB)