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

• Label: 2.37.i_bb
• 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:	$1 + 8 x + 27 x^{2} + 296 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.395093238421$, $\pm0.938240094913$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{7})\)
Galois group:	$C_2^2$
Jacobians:	$46$
Isomorphism classes:	82
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1701$	$1859193$	$2604060900$	$3507540499449$	$4808085633052821$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$46$	$1360$	$51406$	$1871524$	$69336766$	$2565646270$	$94932463798$	$3512482664644$	$129961731836662$	$4808584285454800$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 46 curves (of which all are hyperelliptic):

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

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

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 \(\Q(\sqrt{-3}, \sqrt{7})\).

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

The base change of $A$ to $\F_{37^{3}}$ is 1.50653.om^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-21}) \)$)$

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.ai_bb	$2$	(not in LMFDB)
2.37.aq_fi	$3$	(not in LMFDB)
2.37.a_k	$6$	(not in LMFDB)