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

• Label: 2.37.r_fo
• 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:	$( 1 + 7 x + 37 x^{2} )( 1 + 10 x + 37 x^{2} )$
	$1 + 17 x + 144 x^{2} + 629 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.695152227498$, $\pm0.807138866923$
Angle rank:	$2$ (numerical)
Jacobians:	$10$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2160$	$1874880$	$2538319680$	$3520312185600$	$4807524706426800$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$55$	$1369$	$50110$	$1878337$	$69328675$	$2565728566$	$94932159895$	$3512478233953$	$129961739383270$	$4808584374285889$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

• $y^2=27 x^6+x^5+28 x^3+15 x+28$
• $y^2=34 x^6+18 x^5+12 x^4+7 x^3+33 x^2+6 x+1$
• $y^2=8 x^6+24 x^5+24 x^4+14 x^3+21 x+23$
• $y^2=30 x^6+28 x^5+15 x^4+18 x^3+34 x^2+15 x+11$
• $y^2=25 x^6+28 x^5+36 x^4+33 x^3+34 x^2+11 x+21$
• $y^2=30 x^6+21 x^5+29 x^4+11 x^3+8 x^2+27 x+14$
• $y^2=7 x^6+27 x^5+3 x^4+28 x^3+7 x^2+34 x+25$
• $y^2=33 x^6+5 x^5+9 x^4+14 x^3+34 x^2+30 x+3$
• $y^2=34 x^6+2 x^5+26 x^4+14 x^3+9 x^2+19 x+3$
• $y^2=15 x^6+6 x^5+11 x^4+16 x^3+27 x^2+17 x+26$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.h $\times$ 1.37.k 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.h : \(\Q(\sqrt{-11}) \).
• 1.37.k : \(\Q(\sqrt{-3}) \).

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.ar_fo	$2$	(not in LMFDB)
2.37.ad_e	$2$	(not in LMFDB)
2.37.d_e	$2$	(not in LMFDB)
2.37.ae_ad	$3$	(not in LMFDB)
2.37.i_dd	$3$	(not in LMFDB)