Invariants
| Base field: | $\F_{79}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 15 x + 79 x^{2} )^{2}$ |
| $1 - 30 x + 383 x^{2} - 2370 x^{3} + 6241 x^{4}$ | |
| Frobenius angles: | $\pm0.180303926787$, $\pm0.180303926787$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $9$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $5, 13$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4225$ | $38130625$ | $243265968400$ | $1517731607705625$ | $9468926435818305625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $50$ | $6108$ | $493400$ | $38966068$ | $3077267750$ | $243089362878$ | $19203920899850$ | $1517108837930788$ | $119851595461800200$ | $9468276072600661548$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=61 x^6+24 x^5+29 x^4+74 x^3+29 x^2+24 x+61$
- $y^2=33 x^6+4 x^5+51 x^4+19 x^3+38 x^2+6 x+62$
- $y^2=57 x^6+49 x^5+31 x^4+29 x^3+31 x^2+49 x+57$
- $y^2=24 x^6+5 x^5+78 x^4+21 x^3+42 x^2+30 x+72$
- $y^2=30 x^6+70 x^5+59 x^4+14 x^3+15 x^2+9 x+33$
- $y^2=31 x^6+41 x^5+3 x^4+73 x^3+3 x^2+41 x+31$
- $y^2=34 x^6+34 x^5+45 x^4+48 x^3+45 x^2+34 x+34$
- $y^2=7 x^6+2 x^5+52 x^4+70 x^3+75 x^2+55 x+5$
- $y^2=28 x^6+75 x^5+52 x^4+40 x^3+52 x^2+75 x+28$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{79}$.
Endomorphism algebra over $\F_{79}$| The isogeny class factors as 1.79.ap 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-91}) \)$)$ |
Base change
This is a primitive isogeny class.