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

• Label: 2.37.ac_cx
• 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:	$( 1 - x + 37 x^{2} )^{2}$
	$1 - 2 x + 75 x^{2} - 74 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.473805533589$, $\pm0.473805533589$
Angle rank:	$1$ (numerical)
Jacobians:	$12$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1369$	$2082249$	$2576983696$	$3502778008041$	$4807660755266209$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$1516$	$50874$	$1868980$	$69330636$	$2565904822$	$94932548460$	$3512473524004$	$129961709026098$	$4808584561055836$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.ab^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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_cv	$2$	(not in LMFDB)
2.37.c_cx	$2$	(not in LMFDB)
2.37.au_gs	$3$	(not in LMFDB)
2.37.al_dg	$3$	(not in LMFDB)
2.37.b_abk	$3$	(not in LMFDB)
2.37.k_cl	$3$	(not in LMFDB)
2.37.w_hn	$3$	(not in LMFDB)