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

• Label: 2.37.a_ab
• Base field: $\F_{37}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{37}$
Dimension:	$2$
L-polynomial:	$1 - x^{2} + 1369 x^{4}$
Frobenius angles:	$\pm0.247849192061$, $\pm0.752150807939$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{3}, \sqrt{-73})\)
Galois group:	$C_2^2$
Jacobians:	$30$
Isomorphism classes:	72
Cyclic group of points:	yes

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1369$	$1874161$	$2565730516$	$3522749856201$	$4808584363053889$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$1368$	$50654$	$1879636$	$69343958$	$2565734622$	$94931877134$	$3512471968228$	$129961739795078$	$4808584353689928$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{37^{2}}$.

Endomorphism algebra over $\F_{37}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{3}, \sqrt{-73})\).

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

The base change of $A$ to $\F_{37^{2}}$ is 1.1369.ab^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-219}) \)$)$

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.a_b	$4$	(not in LMFDB)
2.37.ap_ei	$12$	(not in LMFDB)
2.37.p_ei	$12$	(not in LMFDB)