Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 22 x + 219 x^{2} - 1166 x^{3} + 2809 x^{4}$ |
| Frobenius angles: | $\pm0.101347056399$, $\pm0.310329881205$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-73 -34 \sqrt{2}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $12$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1841$ | $7763497$ | $22210177472$ | $62278904913449$ | $174885069917659801$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $32$ | $2764$ | $149186$ | $7892916$ | $418189752$ | $22164190102$ | $1174710607592$ | $62259702624804$ | $3299763793770362$ | $174887471993889084$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=26 x^6+7 x^5+7 x^3+48 x^2+16 x+32$
- $y^2=33 x^6+7 x^5+3 x^4+31 x^3+4 x^2+37 x+48$
- $y^2=2 x^6+45 x^5+35 x^4+38 x^3+32 x^2+30 x+36$
- $y^2=20 x^6+8 x^5+9 x^4+47 x^3+24 x^2+12 x+19$
- $y^2=50 x^6+36 x^5+17 x^4+35 x^3+15 x^2+7 x+26$
- $y^2=12 x^6+32 x^5+5 x^4+42 x^3+40 x^2+17 x+4$
- $y^2=49 x^6+32 x^5+15 x^4+22 x^3+27 x^2+26 x+14$
- $y^2=2 x^6+51 x^5+35 x^4+27 x^3+52 x^2+10 x+20$
- $y^2=44 x^6+39 x^5+38 x^4+35 x^3+4 x^2+4 x+14$
- $y^2=21 x^6+25 x^5+39 x^4+11 x^3+19 x^2+6 x+8$
- $y^2=19 x^6+51 x^5+32 x^4+42 x^3+14 x^2+4 x+51$
- $y^2=27 x^6+12 x^5+17 x^4+34 x^3+19 x^2+50 x+40$
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 \(\Q(\sqrt{-73 -34 \sqrt{2}})\). |
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.w_il | $2$ | (not in LMFDB) |