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

• Label: 2.61.a_aes
• Base field: $\F_{61}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$( 1 - 61 x^{2} )^{2}$
	$1 - 122 x^{2} + 3721 x^{4}$
Frobenius angles:	$0$, $0$, $1$, $1$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{61}) \)
Galois group:	$C_2$
Jacobians:	$11$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3, 5$

This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3600$	$12960000$	$51519920400$	$191501314560000$	$713342909973690000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$62$	$3478$	$226982$	$13830958$	$844596302$	$51519466438$	$3142742836022$	$191707257613918$	$11694146092834142$	$713342908284497398$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):

• $y^2=x^6+44 x^3+28$
• $y^2=x^6+31 x^3+23$
• $y^2=20 x^6+46 x^5+33 x^4+50 x^3+46 x^2+9 x+11$
• $y^2=42 x^6+43 x^5+48 x^4+20 x^2+26 x+49$
• $y^2=x^6+x^3+11$
• $y^2=20 x^6+45 x^5+60 x^4+33 x^3+28 x^2+22 x+38$
• $y^2=52 x^6+51 x^5+59 x^4+40 x^3+31 x^2+16 x+40$
• $y^2=43 x^6+41 x^5+57 x^4+19 x^3+x^2+32 x+19$
• $y^2=x^5+60 x$
• $y^2=45 x^6+50 x^5+23 x^4+33 x^3+25 x^2+54 x+21$
• $y^2=42 x^6+53 x^5+51 x^4+36 x^3+58 x^2+31 x+32$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61^{2}}$.

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q(\sqrt{61}) \) ramified at both real infinite places.

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

The base change of $A$ to $\F_{61^{2}}$ is 1.3721.aes^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $61$ and $\infty$.

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_cj	$3$	(not in LMFDB)
2.61.a_es	$4$	(not in LMFDB)
2.61.a_a	$8$	(not in LMFDB)