Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 24 x + 244 x^{2} - 1272 x^{3} + 2809 x^{4}$ |
| Frobenius angles: | $\pm0.0392816297272$, $\pm0.272275905364$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.223488.3 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
This isogeny class is simple and 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1758$ | $7647300$ | $22153473918$ | $62263245978000$ | $174880736006125998$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $30$ | $2722$ | $148806$ | $7890934$ | $418179390$ | $22164019954$ | $1174707658182$ | $62259668344606$ | $3299763522984318$ | $174887470627870882$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=5 x^6+9 x^5+18 x^4+12 x^3+35 x^2+20 x+28$
- $y^2=23 x^6+11 x^5+37 x^4+41 x^3+18 x^2+35 x+21$
- $y^2=22 x^6+12 x^5+26 x^4+43 x^3+41 x^2+31 x+17$
- $y^2=27 x^6+x^5+49 x^4+3 x^3+14 x^2+18 x+29$
- $y^2=12 x^6+32 x^5+12 x^4+30 x^3+34 x^2+4 x+5$
- $y^2=30 x^6+9 x^5+37 x^4+17 x^3+39 x^2+8 x$
- $y^2=29 x^6+39 x^5+4 x^4+30 x^2+48 x+3$
- $y^2=18 x^6+39 x^5+29 x^4+47 x^3+19 x^2+6 x+20$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{53}$.
Endomorphism algebra over $\F_{53}$| The endomorphism algebra of this simple isogeny class is 4.0.223488.3. |
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.y_jk | $2$ | (not in LMFDB) |