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

• Label: 2.37.am_eg
• 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:	$( 1 - 6 x + 37 x^{2} )^{2}$
	$1 - 12 x + 110 x^{2} - 444 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.335828188403$, $\pm0.335828188403$
Angle rank:	$1$ (numerical)
Jacobians:	$20$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1024$	$1982464$	$2611618816$	$3517335207936$	$4807352209245184$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$26$	$1446$	$51554$	$1876750$	$69326186$	$2565524022$	$94931320370$	$3512483601694$	$129961785281978$	$4808584491871686$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

• $y^2=34 x^6+31 x^4+31 x^2+34$
• $y^2=11 x^6+12 x^5+9 x^3+17 x^2+19 x+6$
• $y^2=21 x^6+30 x^5+31 x^4+19 x^3+31 x^2+30 x+21$
• $y^2=5 x^6+20 x^5+17 x^4+24 x^3+2 x^2+17 x+27$
• $y^2=15 x^6+26 x^5+36 x^4+5 x^3+36 x^2+26 x+15$
• $y^2=14 x^6+20 x^5+20 x^4+15 x^3+19 x^2+31 x+6$
• $y^2=16 x^6+10 x^4+10 x^2+16$
• $y^2=5 x^6+21 x^5+7 x^4+10 x^3+7 x^2+21 x+5$
• $y^2=24 x^6+22 x^5+4 x^4+11 x^3+21 x^2+32 x+34$
• $y^2=32 x^6+33 x^4+33 x^2+32$
• $y^2=14 x^6+25 x^5+13 x^4+18 x^3+13 x^2+25 x+14$
• $y^2=16 x^6+23 x^5+26 x^4+11 x^3+x^2+14 x+22$
• $y^2=11 x^6+16 x^5+36 x^4+33 x^3+36 x^2+16 x+11$
• $y^2=24 x^6+15 x^5+26 x^4+13 x^3+x^2+14 x+8$
• $y^2=5 x^6+35 x^5+35 x^4+24 x^3+35 x^2+35 x+5$
• $y^2=22 x^6+8 x^5+27 x^4+17 x^3+27 x^2+8 x+22$
• $y^2=5 x^6+14 x^4+16 x^3+14 x^2+5$
• $y^2=3 x^6+3 x^5+24 x^4+36 x^3+24 x^2+3 x+3$
• $y^2=10 x^6+32 x^4+32 x^2+10$
• $y^2=23 x^6+29 x^5+18 x^4+36 x^3+4 x^2+30 x+30$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.ag^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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.a_bm	$2$	(not in LMFDB)
2.37.m_eg	$2$	(not in LMFDB)
2.37.g_ab	$3$	(not in LMFDB)