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

• Label: 2.89.ag_acb
• 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:	$1 - 6 x - 53 x^{2} - 534 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.0636556779778$, $\pm0.730322344644$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-5})\)
Galois group:	$C_2^2$
Jacobians:	$14$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7329$	$61629561$	$495030445056$	$3936859933609449$	$31180886324769381249$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$7780$	$702198$	$62746564$	$5583910164$	$496980268846$	$44231345115636$	$3936588698897284$	$350356404245006502$	$31181719941084374500$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

• $y^2=71 x^6+9 x^5+12 x^4+66 x^3+53 x^2+61 x+71$
• $y^2=2 x^6+83 x^5+75 x^4+22 x^3+46 x^2+18 x+2$
• $y^2=52 x^6+12 x^5+82 x^4+x^3+x^2+33 x+52$
• $y^2=27 x^6+86 x^5+81 x^4+2 x^3+56 x^2+76 x+27$
• $y^2=23 x^6+76 x^5+54 x^4+47 x^3+19 x^2+62 x+23$
• $y^2=77 x^6+45 x^5+12 x^4+35 x^3+52 x^2+61 x+77$
• $y^2=20 x^6+81 x^5+67 x^4+18 x^3+51 x^2+39 x+20$
• $y^2=20 x^6+16 x^5+38 x^4+18 x^3+80 x^2+15 x+20$
• $y^2=45 x^6+64 x^5+50 x^4+52 x^3+49 x^2+28 x+45$
• $y^2=32 x^6+72 x^5+87 x^4+45 x^3+29 x^2+31 x+32$
• $y^2=88 x^6+67 x^5+53 x^4+28 x^3+59 x^2+16 x+88$
• $y^2=72 x^6+65 x^5+21 x^4+81 x^3+64 x^2+11 x+72$
• $y^2=85 x^6+65 x^5+4 x^4+88 x^3+64 x^2+85$
• $y^2=8 x^6+82 x^5+27 x^4+80 x^3+4 x^2+55 x+8$

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 \(\Q(\sqrt{-3}, \sqrt{-5})\).

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

The base change of $A$ to $\F_{89^{3}}$ is 1.704969.acbi^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$

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.g_acb	$2$	(not in LMFDB)
2.89.m_ig	$3$	(not in LMFDB)
2.89.am_ig	$6$	(not in LMFDB)