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

• Label: 2.89.a_bb
• 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:	$1 + 27 x^{2} + 7921 x^{4}$
Frobenius angles:	$\pm0.274235028360$, $\pm0.725764971640$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{151}, \sqrt{-205})\)
Galois group:	$C_2^2$
Jacobians:	$84$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7949$	$63186601$	$496980669044$	$3938485606596025$	$31181719937671189829$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$7976$	$704970$	$62772468$	$5584059450$	$496980047126$	$44231334895530$	$3936588599865508$	$350356403707485210$	$31181719945376196056$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 84 curves (of which all are hyperelliptic):

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

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

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 \(\Q(\sqrt{151}, \sqrt{-205})\).

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

The base change of $A$ to $\F_{89^{2}}$ is 1.7921.bb^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-30955}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.89.a_abb	$4$	(not in LMFDB)