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

• Label: 2.37.c_cw
• Base field: $\F_{37}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$( 1 + 37 x^{2} )( 1 + 2 x + 37 x^{2} )$
	$1 + 2 x + 74 x^{2} + 74 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.552568456711$
Angle rank:	$1$ (numerical)
Jacobians:	$20$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1520$	$2079360$	$2554987760$	$3503305728000$	$4809433419839600$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$1514$	$50440$	$1869262$	$69356200$	$2565883226$	$94931313160$	$3512474779678$	$129961762513960$	$4808584499927114$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

• $y^2=20 x^6+26 x^5+28 x^4+17 x^3+36 x^2+9 x+15$
• $y^2=x^6+31 x^5+6 x^4+28 x^3+6 x^2+31 x+1$
• $y^2=35 x^6+28 x^5+26 x^4+33 x^2+30 x+17$
• $y^2=8 x^6+2 x^5+15 x^4+6 x^3+15 x^2+2 x+8$
• $y^2=25 x^6+32 x^5+13 x^4+32 x^3+22 x^2+31 x+28$
• $y^2=31 x^6+x^5+9 x^4+17 x^3+9 x^2+x+31$
• $y^2=11 x^6+18 x^5+16 x^4+27 x^3+7 x^2+2 x+27$
• $y^2=20 x^6+31 x^5+24 x^3+31 x+20$
• $y^2=x^6+7 x^5+8 x^4+26 x^3+22 x^2+9 x+26$
• $y^2=27 x^6+7 x^5+23 x^4+27 x^3+23 x^2+7 x+27$
• $y^2=22 x^6+8 x^5+2 x^4+4 x^3+17 x^2+23 x+15$
• $y^2=10 x^6+24 x^5+35 x^4+3 x^3+35 x^2+24 x+10$
• $y^2=33 x^6+3 x^5+3 x^4+25 x^3+27 x^2+21 x+7$
• $y^2=9 x^5+2 x^4+33 x^3+2 x^2+9 x$
• $y^2=21 x^6+29 x^4+2 x^3+13 x^2+12$
• $y^2=11 x^6+22 x^5+9 x^4+15 x^3+9 x^2+22 x+11$
• $y^2=19 x^6+4 x^5+36 x^4+5 x^3+27 x^2+30 x+19$
• $y^2=32 x^6+32 x^5+26 x^4+19 x^3+26 x^2+32 x+32$
• $y^2=5 x^6+15 x^5+29 x^4+2 x^3+20 x^2+29 x+19$
• $y^2=29 x^6+13 x^5+33 x^4+15 x^3+33 x^2+13 x+29$

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.a $\times$ 1.37.c 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.a : \(\Q(\sqrt{-37}) \).
• 1.37.c : \(\Q(\sqrt{-1}) \).

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

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

• 1.1369.cs : \(\Q(\sqrt{-1}) \).
• 1.1369.cw : the quaternion algebra over \(\Q\) ramified at $37$ and $\infty$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.37.ac_cw	$2$	(not in LMFDB)
2.37.am_cw	$4$	(not in LMFDB)
2.37.m_cw	$4$	(not in LMFDB)