Invariants
| Base field: | $\F_{79}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 4 x + 79 x^{2} )( 1 + 4 x + 79 x^{2} )$ |
| $1 + 142 x^{2} + 6241 x^{4}$ | |
| Frobenius angles: | $\pm0.427756044762$, $\pm0.572243955238$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $461$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6384$ | $40755456$ | $243087660144$ | $1516510517760000$ | $9468276078667841904$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $80$ | $6526$ | $493040$ | $38934718$ | $3077056400$ | $243087864766$ | $19203908986160$ | $1517108847680638$ | $119851595982618320$ | $9468276074708836606$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 461 curves (of which all are hyperelliptic):
- $y^2=18 x^6+38 x^4+46 x^3+16 x^2+52 x+14$
- $y^2=54 x^6+35 x^4+59 x^3+48 x^2+77 x+42$
- $y^2=33 x^6+9 x^5+25 x^4+45 x^3+32 x^2+26 x+69$
- $y^2=20 x^6+27 x^5+75 x^4+56 x^3+17 x^2+78 x+49$
- $y^2=29 x^6+32 x^5+55 x^4+60 x^3+60 x^2+63 x+64$
- $y^2=8 x^6+17 x^5+7 x^4+22 x^3+22 x^2+31 x+34$
- $y^2=42 x^6+56 x^5+3 x^4+7 x^3+59 x^2+30 x+38$
- $y^2=47 x^6+10 x^5+9 x^4+21 x^3+19 x^2+11 x+35$
- $y^2=25 x^6+54 x^5+61 x^4+58 x^3+48 x^2+39 x+31$
- $y^2=66 x^6+75 x^5+28 x^4+68 x^3+32 x^2+49 x+1$
- $y^2=40 x^6+67 x^5+5 x^4+46 x^3+17 x^2+68 x+3$
- $y^2=61 x^6+58 x^5+49 x^4+13 x^3+17 x^2+45 x+13$
- $y^2=25 x^6+16 x^5+68 x^4+39 x^3+51 x^2+56 x+39$
- $y^2=45 x^6+59 x^5+47 x^4+47 x^3+72 x^2+16 x+57$
- $y^2=56 x^6+19 x^5+62 x^4+62 x^3+58 x^2+48 x+13$
- $y^2=73 x^6+35 x^4+26 x^2+75$
- $y^2=8 x^6+62 x^4+28 x^2+58$
- $y^2=31 x^6+75 x^5+49 x^4+17 x^3+42 x^2+27 x+4$
- $y^2=14 x^6+67 x^5+68 x^4+51 x^3+47 x^2+2 x+12$
- $y^2=38 x^6+64 x^5+34 x^4+29 x^3+64 x^2+61 x+68$
- and 441 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{79^{2}}$.
Endomorphism algebra over $\F_{79}$| The isogeny class factors as 1.79.ae $\times$ 1.79.e and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
| The base change of $A$ to $\F_{79^{2}}$ is 1.6241.fm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.