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

• Label: 2.37.u_gs
• 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:	$( 1 + 10 x + 37 x^{2} )^{2}$
	$1 + 20 x + 174 x^{2} + 740 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.807138866923$, $\pm0.807138866923$
Angle rank:	$1$ (numerical)
Jacobians:	$8$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2304$	$1806336$	$2554695936$	$3520216498176$	$4806289499998464$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$58$	$1318$	$50434$	$1878286$	$69310858$	$2565904822$	$94931317714$	$3512478446878$	$129961770564058$	$4808584101988678$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

• $y^2=31 x^5+36 x^4+6 x^3+36 x^2+31 x$
• $y^2=21 x^6+21 x^5+10 x^4+10 x^2+21 x+21$
• $y^2=26 x^6+34 x^4+34 x^2+26$
• $y^2=33 x^6+28 x^4+28 x^2+33$
• $y^2=x^6+36$
• $y^2=x^6+x^3+11$
• $y^2=30 x^6+11 x^5+32 x^4+19 x^3+22 x^2+25 x+33$
• $y^2=27 x^6+32 x^5+36 x^4+33 x^3+36 x^2+32 x+27$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.k^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.au_gs	$2$	(not in LMFDB)
2.37.a_aba	$2$	(not in LMFDB)
2.37.aw_hn	$3$	(not in LMFDB)
2.37.ak_cl	$3$	(not in LMFDB)
2.37.ab_abk	$3$	(not in LMFDB)
2.37.c_cx	$3$	(not in LMFDB)
2.37.l_dg	$3$	(not in LMFDB)