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

• Label: 2.17.ac_bi
• Base field: $\F_{17}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$( 1 - 2 x + 17 x^{2} )( 1 + 17 x^{2} )$
	$1 - 2 x + 34 x^{2} - 34 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.422020869623$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$16$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$288$	$103680$	$24609312$	$6900940800$	$2012813418528$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$354$	$5008$	$82622$	$1417616$	$24148386$	$410378768$	$6975653758$	$118587321616$	$2015994553314$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

• $y^2=x^6+x^5+16 x^4+6 x^3+16 x^2+x+1$
• $y^2=2 x^6+2 x^5+12 x^4+9 x^3+12 x^2+2 x+2$
• $y^2=9 x^6+16 x^5+13 x^4+13 x^3+15 x^2+4 x+16$
• $y^2=4 x^6+8 x^5+11 x^4+16 x^3+11 x^2+8 x+4$
• $y^2=14 x^6+2 x^5+x^4+10 x^3+16 x^2+2 x+3$
• $y^2=6 x^6+7 x^5+3 x^4+x^3+3 x^2+7 x+6$
• $y^2=11 x^6+8 x^5+7 x^4+7 x^3+11 x^2+9 x+7$
• $y^2=16 x^6+4 x^5+13 x^4+8 x^3+13 x^2+4 x+16$
• $y^2=9 x^6+5 x^5+6 x^4+7 x^2+12 x+15$
• $y^2=6 x^6+7 x^5+14 x^4+11 x^3+14 x^2+7 x+6$
• $y^2=10 x^6+13 x^5+4 x^4+9 x^2+x+5$
• $y^2=4 x^6+11 x^5+9 x^4+16 x^3+9 x^2+11 x+4$
• $y^2=5 x^6+5 x^5+12 x^4+12 x^3+6 x^2+14 x+7$
• $y^2=13 x^6+8 x^5+2 x^4+12 x^3+2 x^2+8 x+13$
• $y^2=x^6+14 x^5+12 x^4+3 x^3+12 x^2+14 x+1$
• $y^2=15 x^6+5 x^5+4 x^4+5 x^3+4 x^2+5 x+15$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.ac $\times$ 1.17.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.17.ac : \(\Q(\sqrt{-1}) \).
• 1.17.a : \(\Q(\sqrt{-17}) \).

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

The base change of $A$ to $\F_{17^{2}}$ is 1.289.be $\times$ 1.289.bi. The endomorphism algebra for each factor is:

• 1.289.be : \(\Q(\sqrt{-1}) \).
• 1.289.bi : the quaternion algebra over \(\Q\) ramified at $17$ and $\infty$.

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
2.17.c_bi	$2$	(not in LMFDB)
2.17.ai_bi	$4$	(not in LMFDB)
2.17.i_bi	$4$	(not in LMFDB)