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

• Label: 2.37.a_cv
• 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:	$( 1 - x + 37 x^{2} )( 1 + x + 37 x^{2} )$
	$1 + 73 x^{2} + 1369 x^{4}$
Frobenius angles:	$\pm0.473805533589$, $\pm0.526194466411$
Angle rank:	$1$ (numerical)
Jacobians:	$14$
Isomorphism classes:	26

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1443$	$2082249$	$2565815616$	$3502778008041$	$4808584466736843$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$1516$	$50654$	$1868980$	$69343958$	$2565904822$	$94931877134$	$3512473524004$	$129961739795078$	$4808584561055836$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{37^{2}}$.

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.ab $\times$ 1.37.b 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.ab : \(\Q(\sqrt{-3}) \).
• 1.37.b : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{37^{2}}$ is 1.1369.cv^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.ac_cx	$2$	(not in LMFDB)
2.37.c_cx	$2$	(not in LMFDB)
2.37.av_hc	$3$	(not in LMFDB)
2.37.am_dh	$3$	(not in LMFDB)
2.37.aj_cm	$3$	(not in LMFDB)
2.37.a_abv	$3$	(not in LMFDB)
2.37.a_aba	$3$	(not in LMFDB)
2.37.j_cm	$3$	(not in LMFDB)
2.37.m_dh	$3$	(not in LMFDB)
2.37.v_hc	$3$	(not in LMFDB)