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

• Label: 2.37.ap_eu
• 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:	$( 1 - 10 x + 37 x^{2} )( 1 - 5 x + 37 x^{2} )$
	$1 - 15 x + 124 x^{2} - 555 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.192861133077$, $\pm0.365180502153$
Angle rank:	$2$ (numerical)
Jacobians:	$20$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$924$	$1907136$	$2593228176$	$3516980011776$	$4808745500384364$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$23$	$1393$	$51194$	$1876561$	$69346283$	$2565732022$	$94932265199$	$3512482585153$	$129961738575218$	$4808584173540553$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.ak $\times$ 1.37.af 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.ak : \(\Q(\sqrt{-3}) \).
• 1.37.af : \(\Q(\sqrt{-123}) \).

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.af_y	$2$	(not in LMFDB)
2.37.f_y	$2$	(not in LMFDB)
2.37.p_eu	$2$	(not in LMFDB)
2.37.ag_db	$3$	(not in LMFDB)
2.37.g_t	$3$	(not in LMFDB)