Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 102 x^{2} + 2809 x^{4}$ |
| Frobenius angles: | $\pm0.0438619005836$, $\pm0.956138099416$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{13})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $9$ |
| Isomorphism classes: | 22 |
This isogeny class is simple but not geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2708$ | $7333264$ | $22164159476$ | $62184201404416$ | $174887470205226068$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $54$ | $2606$ | $148878$ | $7880910$ | $418195494$ | $22163957822$ | $1174711139838$ | $62259676161694$ | $3299763591802134$ | $174887470044939086$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=36 x^6+44 x^5+46 x^4+37 x^2+7 x+11$
- $y^2=x^6+30 x^5+14 x^4+30 x^2+18 x+29$
- $y^2=19 x^6+33 x^5+4 x^4+5 x^3+34 x^2+39 x+15$
- $y^2=42 x^6+x^5+36 x^4+2 x^2+9 x+21$
- $y^2=5 x^6+49 x^5+x^4+28 x^2+9 x+50$
- $y^2=47 x^6+39 x^5+10 x^4+11 x^2+18 x+43$
- $y^2=33 x^6+22 x^5+19 x^4+48 x^2+2 x+51$
- $y^2=37 x^6+12 x^5+42 x^4+10 x^3+41 x^2+41 x+3$
- $y^2=36 x^6+48 x^5+26 x^4+8 x^3+16 x^2+21 x+26$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{53^{2}}$.
Endomorphism algebra over $\F_{53}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{13})\). |
| The base change of $A$ to $\F_{53^{2}}$ is 1.2809.ady 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$ |
Base change
This is a primitive isogeny class.