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

• Label: 2.37.f_bm
• 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:	$( 1 - 4 x + 37 x^{2} )( 1 + 9 x + 37 x^{2} )$
	$1 + 5 x + 38 x^{2} + 185 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.393356479550$, $\pm0.765077740875$
Angle rank:	$2$ (numerical)
Jacobians:	$28$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1598$	$1946364$	$2571297056$	$3516347915136$	$4806450686189318$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$43$	$1421$	$50764$	$1876225$	$69313183$	$2565711722$	$94932587779$	$3512479327969$	$129961758268588$	$4808584173475061$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.ae $\times$ 1.37.j 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.ae : \(\Q(\sqrt{-33}) \).
• 1.37.j : \(\Q(\sqrt{-67}) \).

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.an_eg	$2$	(not in LMFDB)
2.37.af_bm	$2$	(not in LMFDB)
2.37.n_eg	$2$	(not in LMFDB)