Abelian variety isogeny class 2.61.ay_jt over $\F_{61}$

• Label: 2.61.ay_jt
• Base field: $\F_{61}$
• 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:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$1 - 24 x + 253 x^{2} - 1464 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.0139265954990$, $\pm0.319406737834$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{13})\)
Galois group:	$C_2^2$
Jacobians:	$8$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2487$	$13586481$	$51519935952$	$191672109482841$	$713280258809833287$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$3652$	$226982$	$13843300$	$844522118$	$51519497542$	$3142736839406$	$191707291767364$	$11694146092834142$	$713342911517653252$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=x^6+2x^3+29$
• $y^2=57x^6+14x^5+11x^4+18x^3+57x^2+19x+30$
• $y^2=4x^6+7x^5+30x^4+35x^3+47x^2+40x+3$
• $y^2=x^6+x^3+26$
• $y^2=x^6+x^3+40$
• $y^2=6x^6+44x^5+42x^4+51x^3+41x^2+42x+6$
• $y^2=2x^6+37x^5+12x^4+3x^3+8x^2+15x+2$
• $y^2=2x^6+4x^3+52$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61^{6}}$.

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{13})\).

Endomorphism algebra over $\overline{\F}_{61}$

The base change of $A$ to $\F_{61^{6}}$ is 1.51520374361.ayyny^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-39}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{61^{2}}$

  The base change of $A$ to $\F_{61^{2}}$ is the simple isogeny class 2.3721.acs_btj and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{13})\).

• Endomorphism algebra over $\F_{61^{3}}$

  The base change of $A$ to $\F_{61^{3}}$ is the simple isogeny class 2.226981.a_ayyny and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{13})\).

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
2.61.a_cs	$3$	(not in LMFDB)
2.61.y_jt	$3$	(not in LMFDB)
2.61.a_cs	$6$	(not in LMFDB)