Invariants
Base field: | $\F_{79}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 17 x + 79 x^{2} )( 1 - 15 x + 79 x^{2} )$ |
$1 - 32 x + 413 x^{2} - 2528 x^{3} + 6241 x^{4}$ | |
Frobenius angles: | $\pm0.0944227114288$, $\pm0.180303926787$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $8$ |
Isomorphism classes: | 12 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4095$ | $37735425$ | $242741182320$ | $1517237930979225$ | $9468571386703767975$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $48$ | $6044$ | $492336$ | $38953396$ | $3077152368$ | $243088613822$ | $19203919188432$ | $1517108879925796$ | $119851596338941584$ | $9468276083673641324$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=11x^6+55x^5+8x^4+4x^3+26x^2+23x+11$
- $y^2=41x^6+33x^5+36x^4+22x^3+8x^2+66x+27$
- $y^2=23x^6+69x^5+22x^4+18x^3+38x^2+57x+40$
- $y^2=22x^6+14x^5+39x^4+9x^3+43x^2+54x+38$
- $y^2=78x^6+67x^5+74x^4+10x^3+60x^2+10x+69$
- $y^2=70x^6+60x^5+13x^4+21x^3+64x^2+71x+29$
- $y^2=19x^6+48x^5+33x^4+77x^3+61x^2+75x+26$
- $y^2=77x^6+35x^5+14x^4+33x^3+6x^2+29x+77$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{79}$.
Endomorphism algebra over $\F_{79}$The isogeny class factors as 1.79.ar $\times$ 1.79.ap and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.