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

• Label: 2.89.g_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:	$\pm0.269677655356$, $\pm0.936344322022$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8409$	$61629561$	$498938798736$	$3936859933609449$	$31182553568567486649$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$96$	$7780$	$707742$	$62746564$	$5584208736$	$496980268846$	$44231324675424$	$3936588698897284$	$350356403169963918$	$31181719941084374500$

Jacobians and polarizations

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

• $y^2=35 x^6+27 x^5+36 x^4+20 x^3+70 x^2+5 x+35$
• $y^2=6 x^6+71 x^5+47 x^4+66 x^3+49 x^2+54 x+6$
• $y^2=67 x^6+36 x^5+68 x^4+3 x^3+3 x^2+10 x+67$
• $y^2=9 x^6+88 x^5+27 x^4+60 x^3+78 x^2+55 x+9$
• $y^2=69 x^6+50 x^5+73 x^4+52 x^3+57 x^2+8 x+69$
• $y^2=53 x^6+46 x^5+36 x^4+16 x^3+67 x^2+5 x+53$
• $y^2=60 x^6+65 x^5+23 x^4+54 x^3+64 x^2+28 x+60$
• $y^2=60 x^6+48 x^5+25 x^4+54 x^3+62 x^2+45 x+60$
• $y^2=46 x^6+14 x^5+61 x^4+67 x^3+58 x^2+84 x+46$
• $y^2=7 x^6+38 x^5+83 x^4+46 x^3+87 x^2+4 x+7$
• $y^2=59 x^6+52 x^5+77 x^4+39 x^3+79 x^2+35 x+59$
• $y^2=24 x^6+81 x^5+7 x^4+27 x^3+51 x^2+63 x+24$
• $y^2=58 x^6+81 x^5+31 x^4+59 x^3+51 x^2+58$
• $y^2=24 x^6+68 x^5+81 x^4+62 x^3+12 x^2+76 x+24$

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.cbi^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.ag_acb	$2$	(not in LMFDB)
2.89.am_ig	$3$	(not in LMFDB)
2.89.ag_acb	$6$	(not in LMFDB)