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

• Label: 2.89.a_agw
• Base field: $\F_{89}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{89}$
Dimension:	$2$
L-polynomial:	$( 1 - 89 x^{2} )^{2}$
	$1 - 178 x^{2} + 7921 x^{4}$
Frobenius angles:	$0$, $0$, $1$, $1$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{89}) \)
Galois group:	$C_2$
Jacobians:	$14$

This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7744$	$59969536$	$496979881024$	$3934601256960000$	$31181719918798064704$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$7566$	$704970$	$62710558$	$5584059450$	$496978471086$	$44231334895530$	$3936588554733118$	$350356403707485210$	$31181719907629945806$

Jacobians and polarizations

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

• $y^2=25 x^6+4 x^5+67 x^4+8 x^3+33 x^2+9 x+38$
• $y^2=67 x^6+52 x^5+70 x^4+37 x^2+38 x+44$
• $y^2=75 x^6+46 x^5+78 x^4+62 x^3+65 x^2+27 x+57$
• $y^2=88 x^6+17 x^5+45 x^4+49 x^3+37 x^2+87 x+82$
• $y^2=23 x^6+40 x^5+14 x^4+6 x^3+81 x^2+55 x+1$
• $y^2=27 x^5+30 x^3+3 x$
• $y^2=42 x^6+50 x^5+43 x^4+3 x^3+72 x^2+34 x+46$
• $y^2=27 x^5+81 x^4+8 x^2+62 x$
• $y^2=56 x^6+19 x^5+13 x^4+56 x^3+30 x^2+60 x+44$
• $y^2=79 x^6+57 x^5+39 x^4+79 x^3+x^2+2 x+43$
• $y^2=72 x^6+21 x^5+59 x^4+53 x^3+17 x^2+64 x+74$
• $y^2=53 x^6+49 x^5+57 x^4+25 x^3+56 x^2+11 x+65$
• $y^2=22 x^6+11 x^5+50 x^4+27 x^3+48 x^2+53 x+27$
• $y^2=10 x^6+61 x^5+31 x^4+54 x^3+49 x^2+76 x+26$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{89^{2}}$.

Endomorphism algebra over $\F_{89}$

The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q(\sqrt{89}) \) ramified at both real infinite places.

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

The base change of $A$ to $\F_{89^{2}}$ is 1.7921.agw^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $89$ and $\infty$.

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.a_dl	$3$	(not in LMFDB)
2.89.a_gw	$4$	(not in LMFDB)
2.89.a_dl	$6$	(not in LMFDB)