Abelian variety isogeny class 2.89.j_ai over $\F_{89}$

• Label: 2.89.j_ai
• Base field: $\F_{89}$
• 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:	$\F_{89}$
Dimension:	$2$
L-polynomial:	$1 + 9 x - 8 x^{2} + 801 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.324942153927$, $\pm0.991608820593$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-11})\)
Galois group:	$C_2^2$
Jacobians:	$15$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8724$	$61975296$	$499345742736$	$3936185163500544$	$31182228579070571124$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$99$	$7825$	$708318$	$62735809$	$5584150539$	$496978506286$	$44231339319651$	$3936588721601089$	$350356406008796622$	$31181719927095270625$

Jacobians and polarizations

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

• $y^2=26 x^6+85 x^5+41 x^4+71 x^3+43 x^2+65 x+65$
• $y^2=13 x^6+87 x^5+5 x^4+16 x^3+16 x^2+16 x+17$
• $y^2=75 x^6+63 x^5+69 x^4+65 x^3+30 x^2+66 x+29$
• $y^2=12 x^6+88 x^5+10 x^4+61 x^3+88 x^2+84 x+54$
• $y^2=47 x^6+44 x^5+59 x^4+12 x^3+10 x^2+60 x+47$
• $y^2=41 x^6+39 x^5+52 x^4+80 x^3+56 x+84$
• $y^2=22 x^6+81 x^5+25 x^4+37 x^3+61 x^2+85 x+11$
• $y^2=83 x^6+27 x^5+28 x^4+39 x^3+70 x^2+26 x+83$
• $y^2=53 x^6+22 x^5+65 x^4+16 x^3+38 x^2+29 x+53$
• $y^2=60 x^6+68 x^5+37 x^4+54 x^3+x^2+57 x+71$
• $y^2=87 x^6+80 x^5+17 x^4+69 x^3+33 x^2+5 x+84$
• $y^2=42 x^6+22 x^5+53 x^4+77 x^3+39 x^2+36 x+55$
• $y^2=81 x^6+53 x^5+61 x^4+44 x^3+32 x^2+77 x+81$
• $y^2=73 x^6+22 x^5+59 x^4+26 x^3+65 x^2+60 x+73$
• $y^2=21 x^6+73 x^5+61 x^4+4 x^3+5 x^2+6 x+10$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{89^{3}}$.

Endomorphism algebra over $\F_{89}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-11})\).

Endomorphism algebra over $\overline{\F}_{89}$

The base change of $A$ to $\F_{89^{3}}$ is 1.704969.cmk^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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
2.89.aj_ai	$2$	(not in LMFDB)
2.89.as_jz	$3$	(not in LMFDB)
2.89.a_dt	$6$	(not in LMFDB)