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

• Label: 2.37.ap_ei
• 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:	$1 - 15 x + 112 x^{2} - 555 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.0811825253941$, $\pm0.414515858727$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{73})\)
Galois group:	$C_2^2$
Jacobians:	$28$
Isomorphism classes:	28
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$912$	$1871424$	$2565722304$	$3507355489536$	$4807123165898832$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$23$	$1369$	$50654$	$1871425$	$69322883$	$2565718198$	$94932595319$	$3512483196769$	$129961739795078$	$4808584363053889$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{37^{6}}$.

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 $\F_{37^{6}}$ is 1.2565726409.agby^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-219}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{37^{2}}$

  The base change of $A$ to $\F_{37^{2}}$ is the simple isogeny class 2.1369.ab_acaq and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{73})\).

• Endomorphism algebra over $\F_{37^{3}}$

  The base change of $A$ to $\F_{37^{3}}$ is the simple isogeny class 2.50653.a_agby and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{73})\).

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.p_ei	$2$	(not in LMFDB)
2.37.a_b	$3$	(not in LMFDB)
2.37.p_ei	$3$	(not in LMFDB)