Invariants
| Base field: | $\F_{79}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 16 x + 79 x^{2} )^{2}$ |
| $1 - 32 x + 414 x^{2} - 2528 x^{3} + 6241 x^{4}$ | |
| Frobenius angles: | $\pm0.143514932644$, $\pm0.143514932644$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $14$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4096$ | $37748736$ | $242788765696$ | $1517333092761600$ | $9468707275449143296$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $48$ | $6046$ | $492432$ | $38955838$ | $3077196528$ | $243089242846$ | $19203926513232$ | $1517108949141118$ | $119851596825732528$ | $9468276085117144606$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=33x^6+70x^5+71x^4+12x^3+71x^2+70x+33$
- $y^2=44x^6+46x^5+49x^4+13x^3+49x^2+46x+44$
- $y^2=77x^6+34x^4+34x^2+77$
- $y^2=25x^6+24x^5+71x^4+62x^3+58x^2+37x+16$
- $y^2=78x^6+29x^4+29x^2+78$
- $y^2=3x^6+3x^3+77$
- $y^2=3x^6+12x^3+34$
- $y^2=14x^6+49x^5+11x^4+77x^3+11x^2+49x+14$
- $y^2=35x^6+77x^5+3x^4+2x^3+3x^2+77x+35$
- $y^2=30x^6+19x^5+48x^4+36x^3+48x^2+19x+30$
- $y^2=74x^6+22x^5+39x^4+29x^3+75x^2+65x+47$
- $y^2=66x^6+10x^4+10x^2+66$
- $y^2=25x^6+28x^5+64x^3+28x+25$
- $y^2=62x^6+30x^4+30x^2+62$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{79}$.
Endomorphism algebra over $\F_{79}$| The isogeny class factors as 1.79.aq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$ |
Base change
This is a primitive isogeny class.