Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 22 x + 224 x^{2} - 1166 x^{3} + 2809 x^{4}$ |
| Frobenius angles: | $\pm0.161228510555$, $\pm0.280371056196$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.906048.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1846$ | $7793812$ | $22259616262$ | $62320349535184$ | $174906691046785126$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $32$ | $2774$ | $149516$ | $7898166$ | $418241452$ | $22164496022$ | $1174710980272$ | $62259688687774$ | $3299763617814032$ | $174887470758892934$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=12 x^6+33 x^5+5 x^4+18 x^3+31 x^2+12 x+20$
- $y^2=43 x^6+33 x^5+37 x^4+9 x^3+13 x^2+33 x+27$
- $y^2=7 x^6+20 x^5+47 x^4+32 x^3+39 x^2+41 x+19$
- $y^2=22 x^6+2 x^5+23 x^3+24 x^2+39 x+48$
- $y^2=50 x^6+39 x^5+24 x^4+29 x^3+17 x^2+2 x+2$
- $y^2=43 x^6+41 x^5+10 x^4+41 x^3+28 x^2+45 x+28$
- $y^2=43 x^6+44 x^5+2 x^4+24 x^3+43 x^2+13 x+26$
- $y^2=51 x^6+41 x^5+46 x^4+50 x^3+47 x^2+24 x+14$
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 4.0.906048.1. |
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.w_iq | $2$ | (not in LMFDB) |