Invariants
| Base field: | $\F_{79}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 7 x + 79 x^{2} )^{2}$ |
| $1 + 14 x + 207 x^{2} + 1106 x^{3} + 6241 x^{4}$ | |
| Frobenius angles: | $\pm0.628833084391$, $\pm0.628833084391$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $24$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3, 29$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $7569$ | $40335201$ | $241792492176$ | $1517155706166489$ | $9468890014563008649$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $94$ | $6460$ | $490408$ | $38951284$ | $3077255914$ | $243085963966$ | $19203903665446$ | $1517108964984484$ | $119851595317409272$ | $9468276075032154700$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=39 x^6+6 x^5+48 x^4+5 x^3+3 x^2+12 x+69$
- $y^2=x^6+2 x^3+8$
- $y^2=42 x^6+43 x^5+54 x^4+30 x^3+51 x^2+37 x+43$
- $y^2=65 x^6+38 x^5+41 x^4+16 x^3+41 x^2+38 x+65$
- $y^2=29 x^6+77 x^5+44 x^4+26 x^3+59 x^2+2 x+40$
- $y^2=42 x^6+74 x^5+18 x^4+45 x^3+59 x^2+43 x+38$
- $y^2=22 x^6+30 x^5+49 x^4+73 x^3+72 x^2+14 x+26$
- $y^2=15 x^6+46 x^5+54 x^4+11 x^3+54 x^2+46 x+15$
- $y^2=10 x^6+12 x^5+16 x^4+64 x^3+75 x^2+14 x+9$
- $y^2=37 x^6+66 x^5+38 x^4+60 x^3+38 x^2+66 x+37$
- $y^2=20 x^6+78 x^5+71 x^4+41 x^3+65 x^2+21 x+5$
- $y^2=54 x^6+36 x^5+60 x^4+70 x^3+35 x^2+23 x+78$
- $y^2=74 x^6+12 x^5+70 x^4+54 x^3+34 x^2+15 x+42$
- $y^2=6 x^6+39 x^5+18 x^4+21 x^3+19 x^2+37 x+12$
- $y^2=23 x^6+59 x^5+19 x^4+46 x^3+3 x^2+18 x+31$
- $y^2=17 x^6+37 x^5+58 x^4+3 x^3+46 x^2+61 x+9$
- $y^2=51 x^6+47 x^5+55 x^4+15 x^3+26 x^2+40 x+76$
- $y^2=60 x^6+13 x^5+20 x^4+6 x^3+20 x^2+13 x+60$
- $y^2=9 x^6+22 x^5+3 x^4+53 x^3+47 x^2+58 x+31$
- $y^2=35 x^6+73 x^5+44 x^4+4 x^3+16 x^2+26 x+32$
- $y^2=45 x^6+69 x^5+46 x^4+51 x^3+46 x^2+53 x+23$
- $y^2=72 x^6+18 x^5+78 x^4+27 x^3+69 x+17$
- $y^2=27 x^6+49 x^5+32 x^4+35 x^3+23 x^2+39 x+38$
- $y^2=12 x^6+6 x^5+58 x^4+64 x^3+22 x^2+71 x+12$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{79}$.
Endomorphism algebra over $\F_{79}$| The isogeny class factors as 1.79.h 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-267}) \)$)$ |
Base change
This is a primitive isogeny class.