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

• Label: 2.37.e_da
• 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:	$( 1 + 2 x + 37 x^{2} )^{2}$
	$1 + 4 x + 78 x^{2} + 148 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.552568456711$, $\pm0.552568456711$
Angle rank:	$1$ (numerical)
Jacobians:	$49$
Cyclic group of points:	no
Non-cyclic primes:	$2, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1600$	$2073600$	$2544193600$	$3504384000000$	$4810282478440000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$1510$	$50226$	$1869838$	$69368442$	$2565837430$	$94930749186$	$3512477602078$	$129961785232842$	$4808584350060550$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 49 curves (of which all are hyperelliptic):

• $y^2=2 x^6+13 x^3+22$
• $y^2=3 x^5+12 x^4+2 x^3+21 x^2+19 x+13$
• $y^2=13 x^6+25 x^4+25 x^2+13$
• $y^2=18 x^6+3 x^5+27 x^4+20 x^3+27 x^2+3 x+18$
• $y^2=11 x^6+9 x^5+3 x^4+30 x^3+27 x^2+18 x+7$
• $y^2=16 x^6+4 x^5+34 x^4+6 x^3+34 x^2+4 x+16$
• $y^2=10 x^6+20 x^5+20 x^3+20 x+10$
• $y^2=31 x^6+29 x^5+6 x^4+10 x^3+24 x^2+20 x+23$
• $y^2=11 x^6+30 x^5+x^4+22 x^3+34 x^2+11 x+36$
• $y^2=28 x^6+7 x^4+21 x^3+5 x^2+33 x+17$
• $y^2=14 x^6+20 x^5+13 x^4+30 x^3+2 x^2+32 x+8$
• $y^2=19 x^6+9 x^5+30 x^4+15 x^3+27 x^2+x+13$
• $y^2=13 x^6+10 x^5+23 x^4+7 x^3+29 x^2+x+24$
• $y^2=31 x^5+27 x^4+10 x^3+21 x^2+32 x$
• $y^2=27 x^6+25 x^5+23 x^4+10 x^3+23 x^2+25 x+27$
• $y^2=x^6+9 x^5+9 x^4+27 x^3+8 x^2+8 x+26$
• $y^2=7 x^6+11 x^5+32 x^4+31 x^3+2 x^2+21 x+33$
• $y^2=30 x^6+35 x^5+28 x^4+25 x^3+3 x^2+8 x+3$
• $y^2=33 x^6+36 x^4+36 x^2+33$
• $y^2=18 x^6+35 x^5+30 x^4+30 x^3+30 x^2+35 x+18$
• and

• $y^2=28 x^6+11 x^5+31 x^4+14 x^3+29 x^2+36 x+28$
• $y^2=2 x^6+31 x^3+20$
• $y^2=2 x^6+2 x^3+2$
• $y^2=32 x^6+30 x^5+x^4+17 x^3+33 x^2+10 x+34$
• $y^2=16 x^6+23 x^5+17 x^4+13 x^3+17 x^2+23 x+16$
• $y^2=5 x^6+6 x^5+25 x^4+29 x^3+4 x^2+13 x+19$
• $y^2=7 x^6+10 x^5+13 x^4+22 x^3+3 x^2+30$
• $y^2=32 x^6+17 x^5+18 x^4+3 x^3+22 x^2+19 x+13$
• $y^2=x^6+3 x^5+13 x^4+12 x^3+6 x^2+21 x+26$
• $y^2=13 x^6+27 x^5+25 x^4+3 x^3+21 x^2+11 x+13$
• $y^2=5 x^6+28 x^4+28 x^2+5$
• $y^2=2 x^6+2 x^3+35$
• $y^2=33 x^6+34 x^5+29 x^4+30 x^3+5 x^2+26 x+3$
• $y^2=24 x^6+30 x^5+30 x^4+14 x^3+7 x^2+30 x+13$
• $y^2=31 x^6+9 x^4+9 x^2+31$
• $y^2=25 x^6+21 x^5+x^4+5 x^3+14 x^2+22 x+14$
• $y^2=2 x^6+29 x^3+20$
• $y^2=13 x^6+11 x^5+7 x^4+10 x^3+27 x^2+30 x+32$
• $y^2=20 x^6+25 x^5+13 x^4+24 x^3+23 x^2+36 x+2$
• $y^2=3 x^6+30 x^4+30 x^2+3$
• $y^2=19 x^6+12 x^5+10 x^4+31 x^3+12 x^2+x+13$
• $y^2=30 x^5+9 x^4+x^3+36 x^2+36 x$
• $y^2=19 x^6+7 x^5+24 x^4+5 x^3+24 x^2+7 x+19$
• $y^2=16 x^6+2 x^4+2 x^2+16$
• $y^2=19 x^6+30 x^5+17 x^4+14 x^3+23 x^2+18 x+8$
• $y^2=21 x^6+9 x^5+14 x^4+10 x^3+11 x^2+11 x+6$
• $y^2=9 x^6+26 x^5+32 x^4+5 x^3+23 x^2+7 x+25$
• $y^2=2 x^6+2 x^3+15$
• $y^2=32 x^6+9 x^5+26 x^4+31 x^3+4 x^2+7 x+13$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.c^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.ae_da	$2$	(not in LMFDB)
2.37.a_cs	$2$	(not in LMFDB)
2.37.ac_abh	$3$	(not in LMFDB)