Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 23 x + 234 x^{2} - 1219 x^{3} + 2809 x^{4}$ |
| Frobenius angles: | $\pm0.118578411632$, $\pm0.275507353539$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.997628.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1802$ | $7723372$ | $22212374624$ | $62298201439424$ | $174898811589738322$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $31$ | $2749$ | $149200$ | $7895361$ | $418222611$ | $22164407002$ | $1174710892519$ | $62259693339489$ | $3299763696768016$ | $174887471664210309$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+12x^5+24x^4+23x^3+5x^2+52x+7$
- $y^2=23x^6+51x^5+37x^3+9x^2+24x+35$
- $y^2=47x^6+23x^5+22x^4+23x^3+17x^2+42x+52$
- $y^2=51x^6+20x^5+12x^4+19x^3+19x^2+26x+15$
- $y^2=21x^6+15x^5+14x^4+x^3+34x^2+11x+35$
- $y^2=42x^6+4x^5+27x^4+31x^3+23x^2+39x+25$
- $y^2=28x^6+47x^5+8x^4+32x^3+37x^2+31x+14$
- $y^2=3x^6+42x^5+34x^4+45x^3+4x^2+4x+9$
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 4.0.997628.1. |
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.x_ja | $2$ | (not in LMFDB) |