Abelian variety isogeny class 2.31.a_ack over $\F_{31}$

• Label: 2.31.a_ack
• Base field: $\F_{31}$
• 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:	$\F_{31}$
Dimension:	$2$
L-polynomial:	$( 1 - 31 x^{2} )^{2}$
	$1 - 62 x^{2} + 961 x^{4}$
Frobenius angles:	$0$, $0$, $1$, $1$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{31}) \)
Galois group:	$C_2$
Jacobians:	$4$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$900$	$810000$	$887444100$	$849346560000$	$819628229722500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$838$	$29792$	$919678$	$28629152$	$887384518$	$27512614112$	$852887343358$	$26439622160672$	$819628172464198$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

• $y^2=22 x^6+9 x^5+30 x^4+19 x^3+15 x^2+27 x+30$
• $y^2=25 x^6+12 x^5+14 x^4+28 x^2+14 x+14$
• $y^2=25 x^6+6 x^5+4 x^4+20 x^2+5 x+25$
• $y^2=x^6+29 x^5+26 x^4+26 x^2+2 x+1$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{31^{2}}$.

Endomorphism algebra over $\F_{31}$

The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q(\sqrt{31}) \) ramified at both real infinite places.

Endomorphism algebra over $\overline{\F}_{31}$

The base change of $A$ to $\F_{31^{2}}$ is 1.961.ack^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $31$ 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
2.31.a_bf	$3$	(not in LMFDB)
2.31.a_ck	$4$	(not in LMFDB)
2.31.a_bf	$6$	(not in LMFDB)