Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 4 x + 37 x^{2} )^{2}$ |
| $1 + 8 x + 90 x^{2} + 296 x^{3} + 1369 x^{4}$ | |
| Frobenius angles: | $\pm0.606643520450$, $\pm0.606643520450$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $26$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 3, 7$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1764$ | $2039184$ | $2527475076$ | $3510137143296$ | $4810882012112484$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $46$ | $1486$ | $49894$ | $1872910$ | $69377086$ | $2565640222$ | $94930996150$ | $3512486166814$ | $129961745539918$ | $4808584101061486$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):
- $y^2=32 x^6+6 x^5+3 x^4+9 x^3+3 x^2+6 x+32$
- $y^2=33 x^6+31 x^5+x^4+13 x^3+9 x^2+32 x+7$
- $y^2=19 x^6+5 x^5+x^4+32 x^3+25 x^2+24 x+8$
- $y^2=26 x^6+5 x^5+9 x^4+22 x^3+34 x^2+27 x+34$
- $y^2=31 x^6+12 x^5+5 x^4+27 x^3+9 x^2+8 x+9$
- $y^2=4 x^6+36 x^5+20 x^4+x^3+13 x^2+3 x+7$
- $y^2=15 x^6+35 x^5+15 x^4+17 x^3+31 x^2+13 x+2$
- $y^2=x^6+x^3+10$
- $y^2=36 x^6+31 x^5+15 x^4+32 x^3+15 x^2+31 x+36$
- $y^2=13 x^6+8 x^5+19 x^4+32 x^3+5 x^2+26 x+9$
- $y^2=7 x^6+25 x^5+4 x^4+20 x^3+4 x^2+25 x+7$
- $y^2=x^6+30 x^3+27$
- $y^2=29 x^6+29 x^5+32 x^4+17 x^3+29 x^2+4 x+7$
- $y^2=13 x^6+12 x^5+15 x^4+11 x^3+15 x^2+12 x+13$
- $y^2=28 x^6+20 x^5+32 x^4+3 x^3+26 x^2+10 x+12$
- $y^2=30 x^6+33 x^5+33 x^4+20 x^3+30 x^2+34 x+7$
- $y^2=11 x^6+27 x^4+27 x^2+11$
- $y^2=5 x^6+30 x^5+34 x^4+4 x^3+19 x^2+22 x+33$
- $y^2=30 x^6+18 x^5+25 x^4+9 x^3+11 x^2+12 x+31$
- $y^2=20 x^6+30 x^5+29 x^4+29 x^3+29 x^2+30 x+20$
- $y^2=4 x^6+24 x^5+13 x^4+4 x^3+13 x^2+24 x+4$
- $y^2=30 x^6+14 x^4+14 x^2+30$
- $y^2=31 x^6+31 x^5+17 x^4+x^3+20 x^2+31 x+29$
- $y^2=34 x^6+22 x^5+3 x^4+5 x^3+25 x^2+7 x+7$
- $y^2=x^6+21 x^5+15 x^4+6 x^3+29 x^2+18 x+13$
- $y^2=12 x^6+30 x^5+9 x^4+30 x^3+35 x^2+16 x+32$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{37}$.
Endomorphism algebra over $\F_{37}$| The isogeny class factors as 1.37.e 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-33}) \)$)$ |
Base change
This is a primitive isogeny class.