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

• Label: 2.37.ag_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:	$( 1 - 3 x + 37 x^{2} )^{2}$
	$1 - 6 x + 83 x^{2} - 222 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.420687118444$, $\pm0.420687118444$
Angle rank:	$1$ (numerical)
Jacobians:	$21$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1225$	$2059225$	$2596921600$	$3506911655625$	$4806395848830625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$1500$	$51266$	$1871188$	$69312392$	$2565741750$	$94933091096$	$3512482528228$	$129961704101402$	$4808584151587500$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 21 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.ad^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-139}) \)$)$

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.a_cn	$2$	(not in LMFDB)
2.37.g_df	$2$	(not in LMFDB)
2.37.d_abc	$3$	(not in LMFDB)