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

• Label: 2.37.r_fi
• Base field: $\F_{37}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{37}$
Dimension:	$2$
L-polynomial:	$1 + 17 x + 138 x^{2} + 629 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.653081809589$, $\pm0.884417636813$
Angle rank:	$2$ (numerical)
Number field:	4.0.2365308.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	12

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2154$	$1856748$	$2553679008$	$3513650499264$	$4809080760639474$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$55$	$1357$	$50416$	$1874785$	$69351115$	$2565686266$	$94931505871$	$3512485950625$	$129961696038064$	$4808584508223637$

Jacobians and polarizations

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

• $y^2=12 x^6+15 x^5+12 x^4+10 x^3+14 x^2+13 x+10$
• $y^2=24 x^6+15 x^5+13 x^4+24 x^3+30 x^2+21 x+33$
• $y^2=25 x^6+10 x^5+30 x^4+24 x^3+9 x^2+19 x+15$
• $y^2=14 x^6+5 x^5+2 x^4+25 x^3+19 x^2+18 x+27$
• $y^2=26 x^6+29 x^5+28 x^4+13 x^3+20 x^2+4 x+20$
• $y^2=4 x^6+14 x^5+19 x^4+17 x^3+26 x^2+36 x+36$
• $y^2=23 x^6+6 x^5+17 x^4+18 x^3+17 x^2+25 x+18$
• $y^2=8 x^6+15 x^5+16 x^4+x^3+9 x^2+26 x+16$
• $y^2=4 x^6+18 x^5+28 x^4+26 x^3+21 x^2+3 x+6$
• $y^2=21 x^6+10 x^5+5 x^4+20 x^3+31 x^2+14 x+13$
• $y^2=3 x^6+17 x^5+31 x^4+8 x^3+32 x^2+15 x+10$
• $y^2=36 x^6+35 x^5+26 x^4+5 x^3+9 x^2+3 x+34$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The endomorphism algebra of this simple isogeny class is 4.0.2365308.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.37.ar_fi	$2$	(not in LMFDB)