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 contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=5x^6+9x^5+18x^4+12x^3+35x^2+20x+28$
- $y^2=23x^6+11x^5+37x^4+41x^3+18x^2+35x+21$
- $y^2=22x^6+12x^5+26x^4+43x^3+41x^2+31x+17$
- $y^2=27x^6+x^5+49x^4+3x^3+14x^2+18x+29$
- $y^2=12x^6+32x^5+12x^4+30x^3+34x^2+4x+5$
- $y^2=30x^6+9x^5+37x^4+17x^3+39x^2+8x$
- $y^2=29x^6+39x^5+4x^4+30x^2+48x+3$
- $y^2=18x^6+39x^5+29x^4+47x^3+19x^2+6x+20$
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) |