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

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

Invariants

Base field:	$\F_{37}$
Dimension:	$2$
L-polynomial:	$1 + 5 x - 12 x^{2} + 185 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.301486164513$, $\pm0.968152831180$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{41})\)
Galois group:	$C_2^2$
Jacobians:	$10$
Isomorphism classes:	18

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1548$	$1808064$	$2609575056$	$3511846100736$	$4809570923227068$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$43$	$1321$	$51514$	$1873825$	$69358183$	$2565559222$	$94931768779$	$3512475819169$	$129961768124338$	$4808584436080561$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

• $y^2=11 x^6+18 x^5+18 x^4+11 x^3+18 x^2+25 x$
• $y^2=26 x^6+30 x^5+9 x^4+36 x^3+23 x^2+25 x+18$
• $y^2=8 x^6+24 x^5+9 x^4+3 x^3+4 x^2+x+33$
• $y^2=16 x^6+18 x^5+18 x^4+29 x^3+29 x^2+20 x+18$
• $y^2=20 x^5+6 x^4+33 x^3+29 x+5$
• $y^2=28 x^6+33 x^5+27 x^4+36 x^3+18 x^2+3 x+31$
• $y^2=8 x^6+19 x^5+20 x^4+7 x^3+x+36$
• $y^2=x^6+17 x^5+32 x^4+31 x^3+17 x^2+14 x+25$
• $y^2=x^6+x^3+7$
• $y^2=19 x^6+7 x^5+6 x^4+11 x^3+29 x^2+25 x+14$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{37^{3}}$.

Endomorphism algebra over $\F_{37}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{41})\).

Endomorphism algebra over $\overline{\F}_{37}$

The base change of $A$ to $\F_{37^{3}}$ is 1.50653.qo^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$

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_am	$2$	(not in LMFDB)
2.37.ak_dv	$3$	(not in LMFDB)
2.37.a_bx	$6$	(not in LMFDB)