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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $8409$ | $61629561$ | $498938798736$ | $3936859933609449$ | $31182553568567486649$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $96$ | $7780$ | $707742$ | $62746564$ | $5584208736$ | $496980268846$ | $44231324675424$ | $3936588698897284$ | $350356403169963918$ | $31181719941084374500$ |
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 \(\Q(\sqrt{-3}, \sqrt{-5})\). |
| 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.