Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 101 x^{2} + 2809 x^{4}$ |
| Frobenius angles: | $\pm0.0490782749449$, $\pm0.950921725055$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-5}, \sqrt{23})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $5$ |
| Isomorphism classes: | 26 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2709$ | $7338681$ | $22164181956$ | $62187403038201$ | $174887470341297189$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $54$ | $2608$ | $148878$ | $7881316$ | $418195494$ | $22164002782$ | $1174711139838$ | $62259679965508$ | $3299763591802134$ | $174887470317081328$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=30 x^6+2 x^5+42 x^4+44 x^3+34 x^2+34 x+4$
- $y^2=47 x^6+12 x^5+20 x^4+45 x^3+28 x^2+32 x+20$
- $y^2=41 x^6+25 x^5+17 x^3+44 x+29$
- $y^2=12 x^6+x^5+40 x^4+31 x^3+34 x^2+52 x+11$
- $y^2=36 x^6+23 x^5+28 x^4+48 x^3+3 x^2+39 x+23$
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 \(\Q(\sqrt{-5}, \sqrt{23})\). |
| The base change of $A$ to $\F_{53^{2}}$ is 1.2809.adx 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-115}) \)$)$ |
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_dx | $4$ | (not in LMFDB) |