Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 23 x + 235 x^{2} - 1219 x^{3} + 2809 x^{4}$ |
| Frobenius angles: | $\pm0.133260592692$, $\pm0.268001606122$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-306 -46 \sqrt{13}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $9$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1803$ | $7729461$ | $22222719639$ | $62307502028901$ | $174904246261677648$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $31$ | $2751$ | $149269$ | $7896539$ | $418235606$ | $22164502671$ | $1174711254125$ | $62259691660243$ | $3299763653532823$ | $174887471233559046$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=35 x^6+40 x^5+29 x^4+17 x^3+30 x^2+33 x+14$
- $y^2=31 x^6+23 x^5+18 x^4+2 x^2+11 x+13$
- $y^2=41 x^6+28 x^5+49 x^4+35 x^3+51 x^2+42 x+19$
- $y^2=3 x^6+11 x^5+6 x^4+43 x^3+45 x^2+5 x+26$
- $y^2=52 x^6+2 x^5+50 x^4+21 x^3+36 x^2+46 x+3$
- $y^2=8 x^6+24 x^5+27 x^4+6 x^3+3 x+5$
- $y^2=26 x^6+34 x^5+18 x^4+51 x^3+33 x^2+14 x+35$
- $y^2=5 x^6+14 x^5+6 x^4+20 x^3+4 x^2+15 x+6$
- $y^2=19 x^6+21 x^5+34 x^4+35 x^3+44 x^2+4 x+42$
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 \(\Q(\sqrt{-306 -46 \sqrt{13}})\). |
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_jb | $2$ | (not in LMFDB) |