Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 14 x + 53 x^{2} )( 1 - 11 x + 53 x^{2} )$ |
| $1 - 25 x + 260 x^{2} - 1325 x^{3} + 2809 x^{4}$ | |
| Frobenius angles: | $\pm0.0885855327829$, $\pm0.227402221936$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $6$ |
| Isomorphism classes: | 16 |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1720$ | $7602400$ | $22149554560$ | $62282662000000$ | $174899964521176600$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $29$ | $2705$ | $148778$ | $7893393$ | $418225369$ | $22164513590$ | $1174711321813$ | $62259686728513$ | $3299763570508034$ | $174887470603558025$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=18x^6+46x^5+17x^4+19x^3+38x^2+40x+27$
- $y^2=39x^6+36x^5+30x^4+25x^3+24x^2+52x+8$
- $y^2=24x^6+36x^5+8x^4+21x^3+36x^2+29x+39$
- $y^2=39x^6+11x^5+16x^4+20x^3+38x^2+25x+11$
- $y^2=50x^6+18x^5+9x^4+22x^3+49x^2+6x+3$
- $y^2=45x^6+2x^5+11x^4+42x^3+7x^2+4x+41$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{53}$.
Endomorphism algebra over $\F_{53}$| The isogeny class factors as 1.53.ao $\times$ 1.53.al and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.