Abelian variety isogeny class 2.97.abf_qf over $\F_{97}$

• Label: 2.97.abf_qf
• Base field: $\F_{97}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{97}$
Dimension:	$2$
L-polynomial:	$1 - 31 x + 421 x^{2} - 3007 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.0759227908932$, $\pm0.294331848809$
Angle rank:	$2$ (numerical)
Number field:	4.0.15047813.1
Galois group:	$D_{4}$
Jacobians:	$12$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6793$	$87419117$	$833282480881$	$7837884123110549$	$73741966541689270528$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$67$	$9291$	$913015$	$88534371$	$8587288302$	$832970396883$	$80798268596287$	$7837433574914019$	$760231060390919827$	$73742412718887178366$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=82 x^6+61 x^5+18 x^4+6 x^3+78 x^2+27 x+23$
• $y^2=30 x^6+7 x^5+84 x^4+41 x^3+14 x^2+28 x+67$
• $y^2=63 x^6+5 x^5+24 x^4+11 x^3+69 x^2+91 x+76$
• $y^2=67 x^6+86 x^5+33 x^4+2 x^3+22 x^2+58 x+45$
• $y^2=82 x^6+37 x^5+26 x^4+61 x^3+50 x^2+12 x+75$
• $y^2=21 x^6+11 x^5+19 x^4+6 x^3+39 x^2+85 x+43$
• $y^2=15 x^6+75 x^5+40 x^4+38 x^3+23 x^2+81 x+76$
• $y^2=29 x^6+41 x^5+18 x^4+22 x^3+11 x^2+11 x+5$
• $y^2=65 x^6+54 x^5+13 x^4+35 x^3+23 x^2+55 x+15$
• $y^2=55 x^6+28 x^5+70 x^4+48 x^3+50 x^2+80 x+26$
• $y^2=17 x^6+37 x^5+3 x^4+4 x^3+65 x^2+94 x+80$
• $y^2=94 x^6+25 x^5+28 x^4+38 x^3+57 x^2+87 x+1$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{97}$.

Endomorphism algebra over $\F_{97}$

The endomorphism algebra of this simple isogeny class is 4.0.15047813.1.

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
2.97.bf_qf	$2$	(not in LMFDB)