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

• Label: 2.37.ah_m
• 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:	$1 - 7 x + 12 x^{2} - 259 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.0284855608318$, $\pm0.638181105835$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-11})\)
Galois group:	$C_2^2$
Jacobians:	$23$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1116$	$1839168$	$2522048400$	$3508521940224$	$4808496445884396$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$31$	$1345$	$49786$	$1872049$	$69342691$	$2565552310$	$94931314663$	$3512480170369$	$129961708202482$	$4808584235335225$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 23 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{37^{3}}$.

Endomorphism algebra over $\F_{37}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-11})\).

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

The base change of $A$ to $\F_{37^{3}}$ is 1.50653.aqs^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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.h_m	$2$	(not in LMFDB)
2.37.o_et	$3$	(not in LMFDB)
2.37.ao_et	$6$	(not in LMFDB)