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

• Label: 2.17.ai_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:	$( 1 - 8 x + 17 x^{2} )( 1 + 17 x^{2} )$
	$1 - 8 x + 34 x^{2} - 136 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.0779791303774$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$16$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$180$	$84240$	$23636340$	$6900940800$	$2014849494900$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$294$	$4810$	$82622$	$1419050$	$24146406$	$410344490$	$6975653758$	$118588284490$	$2015998927014$

Jacobians and polarizations

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

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

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.ai $\times$ 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.ai : \(\Q(\sqrt{-1}) \).
• 1.17.a : \(\Q(\sqrt{-17}) \).

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

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

• 1.289.abe : \(\Q(\sqrt{-1}) \).
• 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.i_bi	$2$	(not in LMFDB)
2.17.ac_bi	$4$	(not in LMFDB)
2.17.c_bi	$4$	(not in LMFDB)