Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 13 x + 53 x^{2} )( 1 - 9 x + 53 x^{2} )$ |
| $1 - 22 x + 223 x^{2} - 1166 x^{3} + 2809 x^{4}$ | |
| Frobenius angles: | $\pm0.148706751109$, $\pm0.287893547303$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $30$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1845$ | $7787745$ | $22249725840$ | $62312123007225$ | $174902550714022725$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $32$ | $2772$ | $149450$ | $7897124$ | $418231552$ | $22164446934$ | $1174711124416$ | $62259694324036$ | $3299763680080610$ | $174887471176107732$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=52 x^6+37 x^5+2 x^4+39 x^3+50 x^2+17 x+10$
- $y^2=15 x^6+46 x^5+37 x^4+10 x^3+16 x^2+2 x+26$
- $y^2=37 x^6+24 x^5+37 x^4+11 x^3+15 x^2+36 x+49$
- $y^2=38 x^6+40 x^5+8 x^4+45 x^3+49 x^2+35 x+45$
- $y^2=50 x^6+45 x^5+47 x^4+15 x^3+38 x^2+7 x+41$
- $y^2=27 x^6+18 x^5+10 x^4+2 x^3+11 x^2+47 x+5$
- $y^2=27 x^6+16 x^5+31 x^4+20 x^3+9 x^2+31 x+8$
- $y^2=34 x^6+17 x^5+42 x^4+32 x^3+8 x^2+40 x+6$
- $y^2=x^6+42 x^5+8 x^4+30 x^3+13 x^2+14 x+30$
- $y^2=50 x^6+19 x^5+47 x^4+24 x^3+13 x^2+20 x+18$
- $y^2=41 x^6+24 x^5+14 x^4+6 x^3+49 x^2+3 x+30$
- $y^2=45 x^6+43 x^5+26 x^4+19 x^3+35 x^2+14 x+49$
- $y^2=18 x^6+28 x^5+12 x^4+2 x^3+47 x^2+29 x+15$
- $y^2=32 x^6+2 x^5+22 x^4+21 x^3+25 x^2+35 x+52$
- $y^2=9 x^6+24 x^5+52 x^4+43 x^3+x^2+24 x+44$
- $y^2=24 x^6+38 x^5+44 x^4+49 x^3+43 x^2+7 x+17$
- $y^2=48 x^6+4 x^5+17 x^4+51 x^3+24 x^2+43 x+27$
- $y^2=5 x^6+51 x^5+33 x^4+17 x^3+34 x^2+8 x+34$
- $y^2=43 x^6+26 x^5+40 x^4+24 x^3+45 x^2+32 x+42$
- $y^2=24 x^6+22 x^5+44 x^4+39 x^3+46 x^2+12 x+15$
- $y^2=35 x^6+41 x^5+28 x^4+51 x^3+45 x^2+22 x+51$
- $y^2=48 x^6+32 x^5+8 x^4+22 x^3+3 x^2+31 x+34$
- $y^2=17 x^6+5 x^5+29 x^4+2 x^3+51 x+33$
- $y^2=45 x^6+19 x^5+5 x^4+37 x^3+39 x^2+26 x+34$
- $y^2=34 x^6+43 x^5+35 x^4+29 x^3+45 x^2+17 x+12$
- $y^2=50 x^6+6 x^5+15 x^4+50 x^3+37 x^2+4 x+41$
- $y^2=8 x^6+15 x^5+18 x^4+50 x^3+39 x^2+13 x+48$
- $y^2=19 x^6+11 x^5+29 x^4+27 x^3+52 x^2+6 x+45$
- $y^2=41 x^6+31 x^5+47 x^4+47 x^3+x^2+20 x+3$
- $y^2=17 x^6+18 x^5+49 x^4+34 x^3+33 x^2+3 x+34$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{53}$.
Endomorphism algebra over $\F_{53}$| The isogeny class factors as 1.53.an $\times$ 1.53.aj 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.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.53.ae_al | $2$ | (not in LMFDB) |
| 2.53.e_al | $2$ | (not in LMFDB) |
| 2.53.w_ip | $2$ | (not in LMFDB) |