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

• Label: 2.37.aq_fe
• 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:	$( 1 - 10 x + 37 x^{2} )( 1 - 6 x + 37 x^{2} )$
	$1 - 16 x + 134 x^{2} - 592 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.192861133077$, $\pm0.335828188403$
Angle rank:	$2$ (numerical)
Jacobians:	$32$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$896$	$1892352$	$2594243456$	$3518775558144$	$4809115816136576$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$1382$	$51214$	$1877518$	$69351622$	$2565714422$	$94931878462$	$3512481024286$	$129961747154038$	$4808584296930182$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.ak $\times$ 1.37.ag and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.37.ak : \(\Q(\sqrt{-3}) \).
• 1.37.ag : \(\Q(\sqrt{-7}) \).

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.ae_o	$2$	(not in LMFDB)
2.37.e_o	$2$	(not in LMFDB)
2.37.q_fe	$2$	(not in LMFDB)
2.37.ah_dc	$3$	(not in LMFDB)
2.37.f_i	$3$	(not in LMFDB)