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

• Label: 2.37.o_du
• 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:	$( 1 + 2 x + 37 x^{2} )( 1 + 12 x + 37 x^{2} )$
	$1 + 14 x + 98 x^{2} + 518 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.552568456711$, $\pm0.947431543289$
Angle rank:	$1$ (numerical)
Jacobians:	$23$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2000$	$1872000$	$2574962000$	$3504384000000$	$4810216590050000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$52$	$1370$	$50836$	$1869838$	$69367492$	$2565726410$	$94931561476$	$3512477602078$	$129961764437332$	$4808584372417850$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 23 curves (of which all are hyperelliptic):

• $y^2=10 x^6+34 x^5+11 x^4+3 x^3+31 x^2+30 x+6$
• $y^2=34 x^5+20 x^4+15 x^3+13 x^2+3 x+30$
• $y^2=26 x^6+13 x^5+15 x^4+6 x^3+14 x^2+3 x+14$
• $y^2=34 x^6+12 x^5+6 x^4+23 x^3+6 x^2+12 x+34$
• $y^2=12 x^6+28 x^5+6 x^4+23 x^3+13 x^2+4 x+9$
• $y^2=10 x^6+4 x^5+32 x^4+32 x^2+33 x+10$
• $y^2=23 x^6+35 x^5+35 x^4+3 x^3+14 x^2+5 x+3$
• $y^2=7 x^6+27 x^5+18 x^4+23 x^3+x^2+3 x+29$
• $y^2=10 x^6+35 x^5+35 x^4+26 x^3+13 x^2+30 x+24$
• $y^2=21 x^6+12 x^5+19 x^4+31 x^3+16 x^2+35 x+4$
• $y^2=3 x^6+22 x^5+26 x^4+12 x^3+30 x^2+19 x+7$
• $y^2=4 x^6+16 x^5+31 x^4+20 x^3+23 x^2+22 x+4$
• $y^2=33 x^6+24 x^5+25 x^4+32 x^3+6 x^2+34 x+29$
• $y^2=25 x^6+7 x^5+14 x^4+28 x^3+32 x^2+16 x+1$
• $y^2=26 x^5+4 x^4+29 x^3+9 x+18$
• $y^2=4 x^6+9 x^5+7 x^4+3 x^3+x^2+21 x+20$
• $y^2=24 x^6+31 x^5+9 x^4+12 x^3+6 x^2+13 x+3$
• $y^2=9 x^6+30 x^5+30 x^4+9 x^3+32 x^2+12 x+1$
• $y^2=36 x^6+24 x^5+9 x^4+12 x^3+25 x^2+20 x+13$
• $y^2=7 x^6+20 x^5+20 x^4+24 x^3+29 x^2+18 x+33$
• $y^2=11 x^6+12 x^5+9 x^4+25 x^3+9 x^2+12 x+11$
• $y^2=25 x^6+35 x^5+33 x^4+6 x^3+16 x^2+20 x+17$
• $y^2=17 x^6+23 x^5+x^4+25 x^3+11 x^2+7 x+27$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{37^{4}}$.

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.c $\times$ 1.37.m 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.c : \(\Q(\sqrt{-1}) \).
• 1.37.m : \(\Q(\sqrt{-1}) \).

Endomorphism algebra over $\overline{\F}_{37}$

The base change of $A$ to $\F_{37^{4}}$ is 1.1874161.adfe^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{37^{2}}$
  The base change of $A$ to $\F_{37^{2}}$ is 1.1369.acs $\times$ 1.1369.cs. The endomorphism algebra for each factor is:

  • 1.1369.acs : \(\Q(\sqrt{-1}) \).
  • 1.1369.cs : \(\Q(\sqrt{-1}) \).

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.ao_du	$2$	(not in LMFDB)
2.37.ak_by	$2$	(not in LMFDB)
2.37.k_by	$2$	(not in LMFDB)