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

• Label: 2.37.aq_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:	$\pm0.271573428246$, $\pm0.271573428246$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$900$	$1904400$	$2604060900$	$3522378240000$	$4809582006322500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$1390$	$51406$	$1879438$	$69358342$	$2565646270$	$94930703806$	$3512473032478$	$129961731836662$	$4808584546343950$

Jacobians and polarizations

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

• $y^2=27 x^6+26 x^5+20 x^4+8 x^3+20 x^2+26 x+27$
• $y^2=6 x^6+19 x^5+5 x^4+17 x^3+35 x^2+6 x+23$
• $y^2=19 x^6+27 x^5+29 x^4+12 x^3+29 x^2+27 x+19$
• $y^2=36 x^6+19 x^5+22 x^4+22 x^3+22 x^2+19 x+36$
• $y^2=19 x^6+24 x^4+24 x^2+19$
• $y^2=28 x^6+11 x^5+15 x^4+13 x^3+15 x^2+11 x+28$
• $y^2=24 x^6+29 x^5+27 x^4+3 x^3+9 x^2+32 x+5$
• $y^2=17 x^6+5 x^4+5 x^2+17$
• $y^2=x^6+4 x^3+27$
• $y^2=11 x^6+25 x^5+18 x^4+2 x^3+18 x^2+25 x+11$
• $y^2=27 x^6+3 x^5+25 x^4+23 x^3+11 x+2$
• $y^2=23 x^6+5 x^5+21 x^4+28 x^3+22 x^2+4 x+14$
• $y^2=23 x^6+7 x^5+6 x^4+6 x^3+24 x^2+x+29$
• $y^2=36 x^6+10 x^5+10 x^4+30 x^3+10 x^2+10 x+36$
• $y^2=23 x^6+17 x^5+24 x^4+23 x^3+4 x^2+35 x+22$
• $y^2=6 x^6+7 x^5+5 x^4+19 x^3+5 x^2+7 x+6$
• $y^2=x^6+9 x^3+26$
• $y^2=15 x^6+35 x^5+31 x^4+15 x^3+30 x^2+36 x+15$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.ai^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.a_k	$2$	(not in LMFDB)
2.37.q_fi	$2$	(not in LMFDB)
2.37.i_bb	$3$	(not in LMFDB)