Invariants
| Base field: | $\F_{79}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 9 x + 79 x^{2} )^{2}$ |
| $1 + 18 x + 239 x^{2} + 1422 x^{3} + 6241 x^{4}$ | |
| Frobenius angles: | $\pm0.668983296649$, $\pm0.668983296649$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $21$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $7921$ | $39929761$ | $241705956496$ | $1517619410523225$ | $9468595729943338921$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $98$ | $6396$ | $490232$ | $38963188$ | $3077160278$ | $243085485246$ | $19203918512282$ | $1517108879823268$ | $119851594600804328$ | $9468276089539753356$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 21 curves (of which all are hyperelliptic):
- $y^2=2 x^6+31 x^5+20 x^4+4 x^3+10 x^2+66 x+18$
- $y^2=51 x^6+61 x^5+59 x^4+20 x^3+59 x^2+61 x+51$
- $y^2=76 x^6+27 x^5+21 x^4+19 x^3+14 x^2+24 x+17$
- $y^2=67 x^6+10 x^5+17 x^4+23 x^3+65 x^2+57 x+30$
- $y^2=77 x^6+31 x^5+64 x^4+72 x^3+64 x^2+31 x+77$
- $y^2=25 x^6+67 x^5+14 x^4+54 x^3+77 x^2+56 x+45$
- $y^2=69 x^6+21 x^5+39 x^4+30 x^3+40 x^2+8 x+13$
- $y^2=68 x^6+73 x^5+20 x^4+19 x^3+72 x^2+45 x+72$
- $y^2=51 x^6+55 x^5+9 x^4+34 x^3+40 x^2+15 x+41$
- $y^2=76 x^6+41 x^5+47 x^4+65 x^3+45 x^2+50 x+55$
- $y^2=68 x^6+32 x^5+71 x^4+30 x^3+2 x^2+7 x+64$
- $y^2=8 x^6+38 x^5+16 x^4+5 x^3+7 x^2+27 x+29$
- $y^2=71 x^6+78 x^5+16 x^4+4 x^3+16 x^2+78 x+71$
- $y^2=15 x^6+65 x^5+63 x^4+37 x^3+21 x^2+52 x+74$
- $y^2=73 x^6+64 x^5+38 x^4+34 x^3+62 x^2+28 x+20$
- $y^2=19 x^6+65 x^4+77 x^3+65 x^2+19$
- $y^2=22 x^6+30 x^5+x^4+31 x^3+x^2+75 x+42$
- $y^2=8 x^6+70 x^5+8 x^4+33 x^3+28 x^2+10 x+53$
- $y^2=32 x^6+45 x^5+39 x^4+39 x^3+39 x^2+45 x+32$
- $y^2=37 x^6+62 x^5+69 x^4+33 x^3+31 x^2+61 x+74$
- $y^2=46 x^6+68 x^5+57 x^4+71 x^3+62 x^2+42 x+38$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{79}$.
Endomorphism algebra over $\F_{79}$| The isogeny class factors as 1.79.j 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-235}) \)$)$ |
Base change
This is a primitive isogeny class.