Abelian variety isogeny class 2.17.a_abi over $\F_{17}$

• Label: 2.17.a_abi
• Base field: $\F_{17}$
• 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:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$( 1 - 17 x^{2} )^{2}$
	$1 - 34 x^{2} + 289 x^{4}$
Frobenius angles:	$0$, $0$, $1$, $1$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{17}) \)
Galois group:	$C_2$
Jacobians:	$2$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$256$	$65536$	$24127744$	$6879707136$	$2015991060736$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$18$	$222$	$4914$	$82366$	$1419858$	$24117918$	$410338674$	$6975423358$	$118587876498$	$2015988221022$

Jacobians and polarizations

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

• $y^2=16 x^6+7 x^5+13 x^4+7 x^3+5 x^2+12 x+7$
• $y^2=10 x^5+13 x^3+3 x$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17^{2}}$.

Endomorphism algebra over $\F_{17}$

The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q(\sqrt{17}) \) ramified at both real infinite places.

Endomorphism algebra over $\overline{\F}_{17}$

The base change of $A$ to $\F_{17^{2}}$ is 1.289.abi^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $17$ 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
2.17.a_r	$3$	(not in LMFDB)
2.17.a_bi	$4$	(not in LMFDB)
2.17.a_a	$8$	(not in LMFDB)