Invariants
Base field: | $\F_{31}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 9 x + 31 x^{2} )( 1 - 7 x + 31 x^{2} )$ |
$1 - 16 x + 125 x^{2} - 496 x^{3} + 961 x^{4}$ | |
Frobenius angles: | $\pm0.200429117370$, $\pm0.283620691308$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $7$ |
Isomorphism classes: | 10 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $575$ | $919425$ | $899990000$ | $855956172825$ | $820012691114375$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $16$ | $956$ | $30208$ | $926836$ | $28642576$ | $887516318$ | $27512382256$ | $852889221796$ | $26439615459328$ | $819628279571276$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+14x^5+18x^4+25x^3+2x^2+4x+24$
- $y^2=x^6+12x^5+24x^4+12x^2+3x+4$
- $y^2=18x^6+16x^5+15x^4+3x^3+21x^2+14x+5$
- $y^2=6x^6+16x^5+11x^4+17x^3+15x^2+9x+12$
- $y^2=15x^6+12x^5+20x^4+x^3+20x^2+12x+15$
- $y^2=3x^6+5x^5+19x^4+22x^3+16x^2+2x+17$
- $y^2=29x^6+9x^4+9x^3+2x^2+30$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{31}$.
Endomorphism algebra over $\F_{31}$The isogeny class factors as 1.31.aj $\times$ 1.31.ah 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.