Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 78 x^{2} + 2809 x^{4}$ |
| Frobenius angles: | $\pm0.118391641536$, $\pm0.881608358464$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-7}, \sqrt{46})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $20$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2732$ | $7463824$ | $22164543884$ | $62252352480256$ | $174887471066133932$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $54$ | $2654$ | $148878$ | $7889550$ | $418195494$ | $22164726638$ | $1174711139838$ | $62259721538974$ | $3299763591802134$ | $174887471766754814$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):
- $y^2=40 x^6+17 x^5+36 x^4+30 x^3+22 x^2+16 x+14$
- $y^2=31 x^6+40 x^5+16 x^4+19 x^3+48 x^2+42 x+42$
- $y^2=34 x^6+48 x^5+35 x^4+x^3+37 x^2+34 x+43$
- $y^2=45 x^6+49 x^5+19 x^4+43 x^3+x^2+44 x+13$
- $y^2=37 x^6+45 x^5+38 x^4+33 x^3+2 x^2+35 x+26$
- $y^2=20 x^6+x^5+26 x^4+38 x^3+46 x^2+37 x+25$
- $y^2=43 x^6+11 x^5+48 x^4+36 x^3+27 x^2+38 x+35$
- $y^2=49 x^6+21 x^5+8 x^4+45 x^3+29 x^2+30 x+2$
- $y^2=16 x^6+49 x^5+31 x^4+49 x^3+46 x^2+25 x+26$
- $y^2=36 x^6+30 x^5+29 x^4+13 x^3+35 x^2+50 x+45$
- $y^2=8 x^6+18 x^5+39 x^4+12 x^3+34 x^2+25 x+34$
- $y^2=15 x^6+51 x^5+6 x^4+7 x^3+3 x^2+26 x+35$
- $y^2=40 x^6+2 x^5+27 x^4+27 x^3+25 x^2+18 x+33$
- $y^2=4 x^6+18 x^5+18 x^4+27 x^3+35 x^2+46 x+48$
- $y^2=8 x^6+36 x^5+36 x^4+x^3+17 x^2+39 x+43$
- $y^2=45 x^6+27 x^5+35 x^4+41 x^3+41 x^2+14 x+51$
- $y^2=17 x^6+47 x^5+5 x^4+17 x^3+47 x^2+38 x+2$
- $y^2=47 x^6+43 x^5+26 x^4+31 x^3+31 x^2+14 x+25$
- $y^2=46 x^6+28 x^5+39 x^4+42 x^3+37 x^2+29 x+22$
- $y^2=9 x^6+34 x^5+26 x^4+46 x^3+7 x^2+39 x+2$
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 \(\Q(\sqrt{-7}, \sqrt{46})\). |
| The base change of $A$ to $\F_{53^{2}}$ is 1.2809.ada 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-322}) \)$)$ |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.53.a_da | $4$ | (not in LMFDB) |