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

• Label: 2.37.q_fi
• 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:	$( 1 + 8 x + 37 x^{2} )^{2}$
	$1 + 16 x + 138 x^{2} + 592 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.728426571754$, $\pm0.728426571754$
Angle rank:	$1$ (numerical)
Jacobians:	$18$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2116$	$1904400$	$2527877284$	$3522378240000$	$4807587119338756$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$1390$	$49902$	$1879438$	$69329574$	$2565646270$	$94933050462$	$3512473032478$	$129961747753494$	$4808584546343950$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

• $y^2=32 x^6+13 x^5+10 x^4+4 x^3+10 x^2+13 x+32$
• $y^2=3 x^6+28 x^5+21 x^4+27 x^3+36 x^2+3 x+30$
• $y^2=28 x^6+32 x^5+33 x^4+6 x^3+33 x^2+32 x+28$
• $y^2=35 x^6+x^5+7 x^4+7 x^3+7 x^2+x+35$
• $y^2=x^6+11 x^4+11 x^2+1$
• $y^2=19 x^6+22 x^5+30 x^4+26 x^3+30 x^2+22 x+19$
• $y^2=12 x^6+33 x^5+32 x^4+20 x^3+23 x^2+16 x+21$
• $y^2=27 x^6+21 x^4+21 x^2+27$
• $y^2=2 x^6+8 x^3+17$
• $y^2=22 x^6+13 x^5+36 x^4+4 x^3+36 x^2+13 x+22$
• $y^2=32 x^6+20 x^5+31 x^4+30 x^3+24 x+1$
• $y^2=30 x^6+21 x^5+29 x^4+14 x^3+11 x^2+2 x+7$
• $y^2=30 x^6+22 x^5+3 x^4+3 x^3+12 x^2+19 x+33$
• $y^2=18 x^6+5 x^5+5 x^4+15 x^3+5 x^2+5 x+18$
• $y^2=9 x^6+34 x^5+11 x^4+9 x^3+8 x^2+33 x+7$
• $y^2=3 x^6+22 x^5+21 x^4+28 x^3+21 x^2+22 x+3$
• $y^2=2 x^6+18 x^3+15$
• $y^2=26 x^6+36 x^5+34 x^4+26 x^3+15 x^2+18 x+26$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.i^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-21}) \)$)$

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.aq_fi	$2$	(not in LMFDB)
2.37.a_k	$2$	(not in LMFDB)
2.37.ai_bb	$3$	(not in LMFDB)