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

• Label: 2.37.m_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:	$\pm0.664171811597$, $\pm0.664171811597$
Angle rank:	$1$ (numerical)
Jacobians:	$20$
Cyclic group of points:	no
Non-cyclic primes:	$2, 11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1936$	$1982464$	$2520441616$	$3517335207936$	$4809816970888336$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$50$	$1446$	$49754$	$1876750$	$69361730$	$2565524022$	$94932433898$	$3512483601694$	$129961694308178$	$4808584491871686$

Jacobians and polarizations

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

• $y^2=17 x^6+34 x^4+34 x^2+17$
• $y^2=22 x^6+24 x^5+18 x^3+34 x^2+x+12$
• $y^2=29 x^6+15 x^5+34 x^4+28 x^3+34 x^2+15 x+29$
• $y^2=21 x^6+10 x^5+27 x^4+12 x^3+x^2+27 x+32$
• $y^2=30 x^6+15 x^5+35 x^4+10 x^3+35 x^2+15 x+30$
• $y^2=7 x^6+10 x^5+10 x^4+26 x^3+28 x^2+34 x+3$
• $y^2=8 x^6+5 x^4+5 x^2+8$
• $y^2=21 x^6+29 x^5+22 x^4+5 x^3+22 x^2+29 x+21$
• $y^2=11 x^6+7 x^5+8 x^4+22 x^3+5 x^2+27 x+31$
• $y^2=27 x^6+29 x^4+29 x^2+27$
• $y^2=28 x^6+13 x^5+26 x^4+36 x^3+26 x^2+13 x+28$
• $y^2=32 x^6+9 x^5+15 x^4+22 x^3+2 x^2+28 x+7$
• $y^2=22 x^6+32 x^5+35 x^4+29 x^3+35 x^2+32 x+22$
• $y^2=12 x^6+26 x^5+13 x^4+25 x^3+19 x^2+7 x+4$
• $y^2=10 x^6+33 x^5+33 x^4+11 x^3+33 x^2+33 x+10$
• $y^2=11 x^6+4 x^5+32 x^4+27 x^3+32 x^2+4 x+11$
• $y^2=21 x^6+7 x^4+8 x^3+7 x^2+21$
• $y^2=20 x^6+20 x^5+12 x^4+18 x^3+12 x^2+20 x+20$
• $y^2=20 x^6+27 x^4+27 x^2+20$
• $y^2=9 x^6+21 x^5+36 x^4+35 x^3+8 x^2+23 x+23$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.g^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.am_eg	$2$	(not in LMFDB)
2.37.a_bm	$2$	(not in LMFDB)
2.37.ag_ab	$3$	(not in LMFDB)