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

• Label: 2.37.k_df
• 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:	$( 1 + x + 37 x^{2} )( 1 + 9 x + 37 x^{2} )$
	$1 + 10 x + 83 x^{2} + 370 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.526194466411$, $\pm0.765077740875$
Angle rank:	$2$ (numerical)
Jacobians:	$28$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1833$	$1966809$	$2546608896$	$3512659902921$	$4808060869563393$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$48$	$1436$	$50274$	$1874260$	$69336408$	$2565844022$	$94931811624$	$3512473006564$	$129961776525498$	$4808584403472236$

Jacobians and polarizations

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

• $y^2=30 x^6+27 x^5+10 x^4+22 x^3+33 x^2+29 x+29$
• $y^2=4 x^6+25 x^4+28 x^3+4 x^2+30$
• $y^2=19 x^6+33 x^5+6 x^3+31 x^2+30 x+25$
• $y^2=2 x^6+19 x^5+32 x^4+16 x^3+30 x^2+30 x+3$
• $y^2=7 x^6+30 x^5+30 x^4+12 x^3+25 x^2+27 x+33$
• $y^2=20 x^6+7 x^5+20 x^4+12 x^3+22 x^2+25 x+34$
• $y^2=32 x^6+2 x^5+30 x^4+2 x^3+4 x^2+36 x+22$
• $y^2=33 x^6+12 x^5+5 x^4+26 x^3+7 x^2+26 x+1$
• $y^2=9 x^6+4 x^5+11 x^4+19 x^3+9 x^2+25 x+25$
• $y^2=28 x^6+31 x^5+27 x^4+5 x^3+9 x^2+24 x+12$
• $y^2=12 x^6+32 x^5+31 x^4+31 x^3+24 x^2+34 x+4$
• $y^2=18 x^6+x^5+x^4+13 x^3+7 x^2+32 x+11$
• $y^2=6 x^6+18 x^5+17 x^4+18 x^3+8 x^2+32 x+28$
• $y^2=6 x^6+4 x^5+8 x^4+2 x^3+16 x^2+27 x+12$
• $y^2=10 x^6+13 x^5+2 x^4+33 x^3+36 x+32$
• $y^2=35 x^6+25 x^5+18 x^4+2 x^3+23 x^2+4 x+1$
• $y^2=7 x^6+19 x^5+3 x^4+17 x^3+4 x+27$
• $y^2=5 x^6+16 x^5+22 x^4+31 x^3+3 x^2+25 x+16$
• $y^2=9 x^6+4 x^5+15 x^4+8 x^3+13 x^2+10 x+34$
• $y^2=27 x^6+34 x^5+32 x^4+13 x^3+9 x^2+8 x+14$
• $y^2=35 x^6+26 x^5+24 x^4+9 x^3+10 x^2+12 x+28$
• $y^2=32 x^6+5 x^5+12 x^4+24 x^3+26 x^2+6 x+24$
• $y^2=24 x^6+8 x^5+17 x^4+12 x^3+18 x^2+21 x+10$
• $y^2=2 x^6+20 x^5+32 x^4+5 x^3+35 x^2+18 x+17$
• $y^2=23 x^6+12 x^5+23 x^4+28 x^3+30 x^2+21 x+21$
• $y^2=36 x^6+35 x^5+8 x^4+20 x^3+5 x^2+5 x+27$
• $y^2=34 x^6+21 x^5+11 x^4+14 x^3+19 x^2+26 x+30$
• $y^2=2 x^6+9 x^5+11 x^4+4 x^3+14 x^2+29 x+30$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.b $\times$ 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.b : \(\Q(\sqrt{-3}) \).
• 1.37.j : \(\Q(\sqrt{-67}) \).

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.ak_df	$2$	(not in LMFDB)
2.37.ai_cn	$2$	(not in LMFDB)
2.37.i_cn	$2$	(not in LMFDB)
2.37.ac_az	$3$	(not in LMFDB)
2.37.t_gi	$3$	(not in LMFDB)