Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 24 x + 248 x^{2} - 1272 x^{3} + 2809 x^{4}$ |
| Frobenius angles: | $\pm0.127135022519$, $\pm0.240897870855$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.205056.2 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1762$ | $7671748$ | $22196676946$ | $62304935276304$ | $174908237671086082$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $30$ | $2730$ | $149094$ | $7896214$ | $418245150$ | $22164630810$ | $1174711981830$ | $62259690799774$ | $3299763595859262$ | $174887470680040650$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+45x^5+35x^4+43x^3+33x^2+11x+41$
- $y^2=48x^6+18x^5+38x^4+23x^3+25x^2+8x+22$
- $y^2=18x^6+38x^5+19x^4+30x^3+7x^2+19x+48$
- $y^2=24x^6+52x^5+16x^4+32x^3+8x^2+21x+30$
- $y^2=51x^6+52x^5+48x^4+27x^3+31x^2+32x+34$
- $y^2=41x^6+39x^5+29x^4+29x^3+13x^2+x+30$
- $y^2=3x^6+38x^5+3x^4+46x^3+26x^2+17x+28$
- $y^2=52x^6+6x^5+11x^4+17x^3+7x+15$
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 4.0.205056.2. |
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_jo | $2$ | (not in LMFDB) |