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

• Label: 2.37.o_et
• 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:	$( 1 + 7 x + 37 x^{2} )^{2}$
	$1 + 14 x + 123 x^{2} + 518 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.695152227498$, $\pm0.695152227498$
Angle rank:	$1$ (numerical)
Jacobians:	$16$
Cyclic group of points:	no
Non-cyclic primes:	$3, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2025$	$1946025$	$2522048400$	$3520407875625$	$4808760230300625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$52$	$1420$	$49786$	$1878388$	$69346492$	$2565552310$	$94933002076$	$3512478021028$	$129961708202482$	$4808584646583100$

Jacobians and polarizations

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

• $y^2=33 x^6+23 x^5+20 x^4+3 x^3+16 x^2+14 x+1$
• $y^2=19 x^6+15 x^5+31 x^4+21 x^3+2 x^2+14 x+13$
• $y^2=x^6+33 x^5+26 x^4+3 x^3+34 x^2+26 x+10$
• $y^2=x^6+16 x^3+26$
• $y^2=8 x^6+17 x^5+32 x^4+15 x^3+36 x^2+19 x+5$
• $y^2=31 x^6+20 x^5+20 x^4+8 x^3+8 x^2+34 x+24$
• $y^2=17 x^6+8 x^5+15 x^4+29 x^3+x^2+26 x+36$
• $y^2=3 x^6+30 x^5+26 x^4+34 x^2+27 x+30$
• $y^2=23 x^6+12 x^5+18 x^4+18 x^3+22 x^2+33 x+29$
• $y^2=23 x^6+9 x^5+20 x^4+19 x^3+15 x^2+12 x+23$
• $y^2=17 x^6+11 x^5+6 x^4+32 x^3+32 x^2+21 x+15$
• $y^2=x^6+3 x^3+27$
• $y^2=15 x^6+28 x^5+13 x^4+34 x^3+22 x^2+4 x+2$
• $y^2=x^6+21 x^3+11$
• $y^2=x^6+24 x^5+4 x^4+16 x^3+28 x^2+29 x+10$
• $y^2=20 x^6+33 x^5+26 x^4+10 x^3+30 x^2+10 x+22$

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^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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.ao_et	$2$	(not in LMFDB)
2.37.a_z	$2$	(not in LMFDB)
2.37.ah_m	$3$	(not in LMFDB)