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: | $\pm0.856485067356$, $\pm0.856485067356$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $14$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $9216$ | $37748736$ | $243388302336$ | $1517333092761600$ | $9467844911930704896$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $112$ | $6046$ | $493648$ | $38955838$ | $3076916272$ | $243089242846$ | $19203891459088$ | $1517108949141118$ | $119851595139504112$ | $9468276085117144606$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):
- $y^2=20 x^6+52 x^5+55 x^4+36 x^3+55 x^2+52 x+20$
- $y^2=41 x^6+68 x^5+69 x^4+57 x^3+69 x^2+68 x+41$
- $y^2=52 x^6+64 x^4+64 x^2+52$
- $y^2=75 x^6+72 x^5+55 x^4+28 x^3+16 x^2+32 x+48$
- $y^2=26 x^6+36 x^4+36 x^2+26$
- $y^2=x^6+x^3+52$
- $y^2=x^6+4 x^3+64$
- $y^2=42 x^6+68 x^5+33 x^4+73 x^3+33 x^2+68 x+42$
- $y^2=38 x^6+52 x^5+x^4+27 x^3+x^2+52 x+38$
- $y^2=11 x^6+57 x^5+65 x^4+29 x^3+65 x^2+57 x+11$
- $y^2=51 x^6+60 x^5+13 x^4+36 x^3+25 x^2+48 x+42$
- $y^2=22 x^6+56 x^4+56 x^2+22$
- $y^2=61 x^6+62 x^5+74 x^3+62 x+61$
- $y^2=47 x^6+10 x^4+10 x^2+47$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{79}$.
Endomorphism algebra over $\F_{79}$| The isogeny class factors as 1.79.q 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$ |
Base change
This is a primitive isogeny class.