Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 8 x + 53 x^{2} )^{2}$ |
| $1 + 16 x + 170 x^{2} + 848 x^{3} + 2809 x^{4}$ | |
| Frobenius angles: | $\pm0.685159765542$, $\pm0.685159765542$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $13$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 31$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3844$ | $8133904$ | $21938941924$ | $62320540880896$ | $174895373513223364$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $70$ | $2894$ | $147358$ | $7898190$ | $418214390$ | $22163801438$ | $1174714615886$ | $62259692266654$ | $3299763392729254$ | $174887471859765614$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 13 curves (of which all are hyperelliptic):
- $y^2=7 x^6+7 x^5+35 x^4+35 x^3+46 x^2+14 x+11$
- $y^2=21 x^6+41 x^5+46 x^4+35 x^3+46 x^2+41 x+21$
- $y^2=28 x^6+x^5+15 x^4+37 x^3+14 x^2+27 x+28$
- $y^2=26 x^6+29 x^5+35 x^4+41 x^3+11 x^2+6 x+17$
- $y^2=11 x^6+27 x^5+28 x^4+43 x^3+52 x^2+21 x+36$
- $y^2=11 x^6+14 x^4+14 x^2+11$
- $y^2=26 x^6+32 x^5+6 x^4+48 x^3+38 x^2+13 x+34$
- $y^2=39 x^6+18 x^5+2 x^4+11 x^3+9 x^2+15 x+4$
- $y^2=43 x^6+39 x^5+12 x^4+12 x^3+45 x^2+14 x+18$
- $y^2=41 x^6+21 x^5+38 x^4+34 x^3+38 x^2+21 x+41$
- $y^2=44 x^6+8 x^5+46 x^4+19 x^3+21 x^2+x+7$
- $y^2=20 x^6+31 x^5+41 x^4+8 x^3+44 x^2+46 x+39$
- $y^2=49 x^6+27 x^5+13 x^4+37 x^3+14 x^2+6 x+46$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{53}$.
Endomorphism algebra over $\F_{53}$| The isogeny class factors as 1.53.i 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-37}) \)$)$ |
Base change
This is a primitive isogeny class.