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

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

Invariants

Base field:	$\F_{97}$
Dimension:	$2$
L-polynomial:	$1 - 9 x - 16 x^{2} - 873 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.0156243205644$, $\pm0.682290987231$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-307})\)
Galois group:	$C_2^2$
Jacobians:	$15$
Isomorphism classes:	27

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8512$	$87469312$	$829527494656$	$7836898028811264$	$73741305883819346752$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$89$	$9297$	$908894$	$88523233$	$8587211369$	$832968511422$	$80798281259561$	$7837433453908801$	$760231055501739038$	$73742412688930519857$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which all are hyperelliptic):

• $y^2=10 x^6+14 x^5+36 x^4+92 x^3+32 x^2+38 x+12$
• $y^2=24 x^6+57 x^5+13 x^4+63 x^3+30 x^2+94 x+51$
• $y^2=72 x^5+59 x^4+71 x^3+8 x^2+72 x+20$
• $y^2=56 x^5+37 x^4+69 x^3+9 x^2+41 x$
• $y^2=83 x^6+91 x^5+65 x^4+33 x^3+9 x^2+16 x+7$
• $y^2=41 x^6+3 x^5+83 x^4+69 x^3+26 x^2+38 x+24$
• $y^2=54 x^6+4 x^5+96 x^4+10 x^3+71 x^2+4 x+24$
• $y^2=94 x^6+15 x^5+15 x^4+23 x^3+14 x^2+45 x+16$
• $y^2=27 x^6+39 x^5+65 x^4+71 x^3+89 x^2+23 x+32$
• $y^2=57 x^6+9 x^5+73 x^4+4 x^3+51 x^2+87 x+37$
• $y^2=21 x^6+85 x^5+70 x^4+23 x^3+23 x^2+29 x+10$
• $y^2=71 x^6+10 x^5+94 x^4+90 x^3+18 x^2+68 x+60$
• $y^2=90 x^6+11 x^5+52 x^4+59 x^3+66 x^2+12 x+69$
• $y^2=7 x^6+57 x^5+89 x^4+86 x^3+86 x^2+12 x+56$
• $y^2=20 x^6+69 x^5+17 x^4+58 x^3+31 x^2+14 x+94$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{97^{3}}$.

Endomorphism algebra over $\F_{97}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-307})\).

Endomorphism algebra over $\overline{\F}_{97}$

The base change of $A$ to $\F_{97^{3}}$ is 1.912673.acus^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-307}) \)$)$

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
2.97.j_aq	$2$	(not in LMFDB)
2.97.s_kp	$3$	(not in LMFDB)
2.97.as_kp	$6$	(not in LMFDB)