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

• Label: 2.37.ar_fk
• 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:	$( 1 - 11 x + 37 x^{2} )( 1 - 6 x + 37 x^{2} )$
	$1 - 17 x + 140 x^{2} - 629 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.140472200256$, $\pm0.335828188403$
Angle rank:	$2$ (numerical)
Jacobians:	$16$
Isomorphism classes:	72
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$864$	$1862784$	$2583000576$	$3515900484096$	$4808653975221984$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$1361$	$50994$	$1875985$	$69344961$	$2565714422$	$94932214125$	$3512484996289$	$129961777923018$	$4808584473040361$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

• $y^2=19 x^6+25 x^5+19 x^4+x^3+14 x^2+15 x+19$
• $y^2=5 x^6+x^5+20 x^4+29 x^3+19 x^2+14 x$
• $y^2=4 x^6+35 x^5+24 x^4+23 x^3+9 x^2+3 x$
• $y^2=18 x^6+4 x^5+4 x^4+4 x^3+10 x^2+x+16$
• $y^2=5 x^6+28 x^5+18 x^4+x^3+14 x^2+5 x+26$
• $y^2=27 x^6+2 x^5+7 x^4+26 x^3+31 x^2+16 x+3$
• $y^2=7 x^6+7 x^5+2 x^4+24 x^3+27 x^2+x+32$
• $y^2=26 x^6+9 x^5+33 x^4+35 x^3+20 x^2+24 x+15$
• $y^2=20 x^6+34 x^5+18 x^4+18 x^3+22 x^2+x+35$
• $y^2=11 x^6+15 x^5+4 x^4+7 x^3+16 x^2+3 x+23$
• $y^2=5 x^6+18 x^5+17 x^4+10 x^3+15 x^2+36 x+16$
• $y^2=13 x^6+30 x^5+19 x^4+8 x^3+10 x^2+24 x+29$
• $y^2=14 x^6+17 x^5+2 x^4+34 x^3+30 x^2+35 x+8$
• $y^2=13 x^6+7 x^5+35 x^4+2 x^3+5 x^2+14 x+11$
• $y^2=33 x^6+20 x^5+19 x^4+29 x^3+20 x^2+30 x+5$
• $y^2=15 x^6+12 x^5+x^4+22 x^3+12 x^2+28 x+29$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.al $\times$ 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.al : \(\Q(\sqrt{-3}) \).
• 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.af_i	$2$	(not in LMFDB)
2.37.f_i	$2$	(not in LMFDB)
2.37.r_fk	$2$	(not in LMFDB)
2.37.af_cq	$3$	(not in LMFDB)
2.37.e_o	$3$	(not in LMFDB)