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

• Label: 2.97.ay_lc
• 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:	$1 - 24 x + 288 x^{2} - 2328 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.0805057840767$, $\pm0.419494215923$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{8})\)
Galois group:	$C_2^2$
Jacobians:	$88$
Isomorphism classes:	213
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7346$	$88519300$	$832906294994$	$7835666472490000$	$73740415438593917906$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$9410$	$912602$	$88509318$	$8587107674$	$832972004930$	$80798305663562$	$7837433749213438$	$760231058457614474$	$73742412689492826050$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 88 curves (of which all are hyperelliptic):

• $y^2=42 x^6+88 x^5+37 x^3+7 x^2+68 x+37$
• $y^2=8 x^6+59 x^5+11 x^4+48 x^3+57 x^2+14 x+10$
• $y^2=67 x^6+89 x^5+78 x^4+9 x^3+70 x^2+93 x+86$
• $y^2=68 x^6+34 x^5+64 x^4+73 x^3+44 x^2+27 x+3$
• $y^2=81 x^6+82 x^5+33 x^4+2 x^3+51 x^2+46 x+38$
• $y^2=73 x^6+74 x^5+58 x^4+21 x^3+90 x^2+2 x+83$
• $y^2=85 x^6+24 x^5+88 x^4+94 x^2+91 x+55$
• $y^2=83 x^6+74 x^5+31 x^4+22 x^3+32 x^2+66 x+51$
• $y^2=85 x^6+95 x^5+41 x^4+25 x^3+62 x^2+20 x+88$
• $y^2=x^6+80 x^5+18 x^4+65 x^3+6 x^2+59 x+67$
• $y^2=15 x^6+65 x^5+93 x^4+32 x^3+47 x^2+46 x+23$
• $y^2=83 x^6+x^5+37 x^4+60 x^3+34 x^2+87 x+39$
• $y^2=31 x^6+60 x^5+42 x^4+20 x^3+84 x^2+81 x+82$
• $y^2=65 x^6+96 x^5+60 x^4+51 x^3+69 x^2+53 x+18$
• $y^2=41 x^6+80 x^5+61 x^4+71 x^3+10 x^2+46 x+77$
• $y^2=56 x^6+95 x^5+19 x^4+88 x^3+14 x^2+89 x+6$
• $y^2=x^6+77 x^5+38 x^4+94 x^3+90 x^2+80 x+15$
• $y^2=54 x^6+68 x^5+60 x^4+88 x^3+12 x^2+87 x+66$
• $y^2=81 x^6+13 x^5+14 x^4+4 x^3+59 x^2+95 x+9$
• $y^2=73 x^6+64 x^5+28 x^4+56 x^3+78 x^2+3 x+35$
• and

• $y^2=x^6+75 x^5+43 x^4+48 x^3+6 x^2+50 x+5$
• $y^2=49 x^6+6 x^5+86 x^4+86 x^2+91 x+49$
• $y^2=18 x^6+26 x^5+2 x^4+63 x^3+93 x^2+17 x+1$
• $y^2=23 x^6+16 x^5+48 x^4+7 x^3+28 x^2+51 x+57$
• $y^2=37 x^6+88 x^5+49 x^4+79 x^3+45 x^2+13 x+51$
• $y^2=26 x^6+83 x^5+96 x^4+51 x^3+32 x^2+44$
• $y^2=18 x^6+8 x^5+83 x^4+95 x^3+9 x^2+13 x+83$
• $y^2=87 x^6+94 x^5+41 x^4+44 x^3+58 x^2+74 x+34$
• $y^2=3 x^6+2 x^5+19 x^4+91 x^3+24 x^2+41 x+49$
• $y^2=87 x^6+11 x^5+23 x^4+77 x^3+19 x^2+83 x+21$
• $y^2=x^6+28 x^5+25 x^4+2 x^3+5 x^2+86 x+35$
• $y^2=26 x^6+93 x^5+94 x^4+94 x^3+67 x^2+13 x+57$
• $y^2=88 x^6+20 x^5+23 x^4+84 x^3+81 x^2+87 x+52$
• $y^2=2 x^6+91 x^5+12 x^4+18 x^3+36 x^2+73 x+52$
• $y^2=44 x^6+91 x^5+4 x^4+46 x^3+90 x^2+34 x+23$
• $y^2=21 x^6+31 x^5+20 x^4+36 x^3+85 x^2+35 x+76$
• $y^2=39 x^6+49 x^5+79 x^4+78 x^3+90 x^2+17 x+94$
• $y^2=46 x^6+78 x^5+73 x^4+56 x^3+41 x^2+60 x+9$
• $y^2=47 x^6+61 x^5+41 x^4+5 x^3+63 x^2+82 x+21$
• $y^2=70 x^6+49 x^5+49 x^4+5 x^3+44 x^2+13 x+81$
• $y^2=35 x^6+88 x^5+89 x^4+44 x^3+96 x^2+4 x+71$
• $y^2=19 x^6+71 x^5+10 x^4+92 x^3+63 x^2+64 x+60$
• $y^2=32 x^6+84 x^5+27 x^4+14 x^3+57 x^2+93 x+34$
• $y^2=89 x^6+77 x^5+35 x^4+57 x^3+92 x^2+82 x+63$
• $y^2=5 x^5+72 x$
• $y^2=77 x^6+68 x^5+73 x^4+20 x^3+59 x^2+54 x+7$
• $y^2=12 x^6+17 x^5+66 x^4+82 x^3+73 x^2+42 x+11$
• $y^2=70 x^6+10 x^5+79 x^4+21 x^3+6 x^2+13 x+45$
• $y^2=87 x^6+80 x^5+17 x^4+63 x^3+25 x^2+66 x+23$
• $y^2=71 x^6+4 x^5+17 x^4+31 x^3+23 x^2+24 x+30$
• $y^2=10 x^6+30 x^5+6 x^4+78 x^3+33 x^2+50 x+21$
• $y^2=96 x^6+82 x^5+71 x^4+38 x^3+51 x^2+70 x+39$
• $y^2=7 x^6+53 x^5+62 x^4+90 x^3+55 x^2+17 x+94$
• $y^2=21 x^6+88 x^5+18 x^4+39 x^3+x^2+40 x+13$
• $y^2=84 x^6+87 x^5+70 x^4+33 x^3+50 x^2+25 x+9$
• $y^2=20 x^6+49 x^5+14 x^4+75 x^3+32 x^2+82 x+30$
• $y^2=34 x^6+73 x^5+71 x^4+71 x^3+31 x^2+51 x+25$
• $y^2=52 x^6+3 x^5+49 x^4+49 x^3+68 x^2+56 x+55$
• $y^2=78 x^6+72 x^5+20 x^4+11 x^3+25 x^2+18 x+68$
• $y^2=51 x^6+94 x^5+94 x^4+50 x^3+65 x^2+33 x+19$
• $y^2=66 x^6+40 x^5+29 x^4+96 x^3+62 x^2+x+34$
• $y^2=85 x^6+5 x^5+56 x^4+14 x^3+37 x^2+26 x+24$
• $y^2=42 x^6+35 x^5+61 x^4+62 x^3+38 x^2+59 x+39$
• $y^2=27 x^6+37 x^5+88 x^4+89 x^3+82 x^2+2 x+40$
• $y^2=45 x^6+87 x^5+65 x^4+27 x^3+86 x^2+50 x+13$
• $y^2=80 x^6+22 x^5+28 x^4+28 x^2+75 x+80$
• $y^2=23 x^6+93 x^5+18 x^4+32 x^3+19 x^2+20 x+20$
• $y^2=69 x^6+2 x^5+46 x^4+42 x^3+87 x^2+72 x+51$
• $y^2=37 x^6+29 x^5+85 x^4+2 x^3+5 x^2+87 x+25$
• $y^2=4 x^6+31 x^5+63 x^4+4 x^3+53 x^2+39 x+71$
• $y^2=47 x^6+49 x^5+80 x^4+80 x^3+25 x^2+74 x+91$
• $y^2=35 x^6+66 x^5+67 x^4+20 x^3+81 x^2+67 x+39$
• $y^2=36 x^6+25 x^5+21 x^4+19 x^3+8 x+28$
• $y^2=54 x^6+83 x^5+88 x^4+8 x^3+94 x^2+68 x+89$
• $y^2=32 x^6+28 x^5+64 x^4+8 x^3+27 x^2+3 x+42$
• $y^2=60 x^6+34 x^5+65 x^4+65 x^3+2 x^2+71 x+74$
• $y^2=74 x^6+6 x^5+28 x^4+26 x^3+34 x^2+35 x+69$
• $y^2=67 x^6+13 x^5+56 x^4+79 x^3+34 x^2+96 x+94$
• $y^2=47 x^6+17 x^5+78 x^4+53 x^3+67 x^2+90 x+21$
• $y^2=52 x^6+49 x^5+69 x^4+82 x^3+x^2+25 x+55$
• $y^2=36 x^6+67 x^5+25 x^4+83 x^3+71 x^2+2 x+80$
• $y^2=59 x^6+27 x^5+57 x^4+29 x^3+21 x^2+36 x+65$
• $y^2=76 x^6+40 x^5+24 x^4+6 x^3+60 x^2+55 x+10$
• $y^2=26 x^6+78 x^5+94 x^4+89 x^3+70 x^2+13 x+32$
• $y^2=7 x^6+53 x^5+70 x^4+70 x^2+44 x+7$
• $y^2=77 x^6+77 x^5+57 x^4+6 x^3+87 x^2+19 x+95$
• $y^2=80 x^6+15 x^5+2 x^4+24 x^3+82 x^2+85 x+13$
• $y^2=29 x^6+23 x^5+67 x^4+82 x^3+3 x^2+34 x+45$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{97^{4}}$.

Endomorphism algebra over $\F_{97}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\).

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

The base change of $A$ to $\F_{97^{4}}$ is 1.88529281.aoty^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{97^{2}}$

  The base change of $A$ to $\F_{97^{2}}$ is the simple isogeny class 2.9409.a_aoty and its endomorphism algebra is \(\Q(\zeta_{8})\).

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.y_lc	$2$	(not in LMFDB)
2.97.au_li	$8$	(not in LMFDB)
2.97.a_adq	$8$	(not in LMFDB)