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

• Label: 2.61.abc_me
• Base field: $\F_{61}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$1 - 28 x + 316 x^{2} - 1708 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.0517882731736$, $\pm0.201777330689$
Angle rank:	$2$ (numerical)
Number field:	4.0.35072.1
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2302$	$13291748$	$51399813358$	$191710018256528$	$713358259036137982$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$34$	$3570$	$226450$	$13846038$	$844614474$	$51520478178$	$3142742314858$	$191707295907294$	$11694145888702018$	$713342910071171890$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=25x^6+46x^5+x^4+7x^3+58x^2+18x+59$
• $y^2=36x^6+9x^5+32x^4+54x^3+30x^2+27x+38$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is 4.0.35072.1.

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
2.61.bc_me	$2$	(not in LMFDB)