Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 7 x + 37 x^{2} )^{2}$ |
| $1 + 14 x + 123 x^{2} + 518 x^{3} + 1369 x^{4}$ | |
| Frobenius angles: | $\pm0.695152227498$, $\pm0.695152227498$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $16$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3, 5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2025$ | $1946025$ | $2522048400$ | $3520407875625$ | $4808760230300625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $52$ | $1420$ | $49786$ | $1878388$ | $69346492$ | $2565552310$ | $94933002076$ | $3512478021028$ | $129961708202482$ | $4808584646583100$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):
- $y^2=33 x^6+23 x^5+20 x^4+3 x^3+16 x^2+14 x+1$
- $y^2=19 x^6+15 x^5+31 x^4+21 x^3+2 x^2+14 x+13$
- $y^2=x^6+33 x^5+26 x^4+3 x^3+34 x^2+26 x+10$
- $y^2=x^6+16 x^3+26$
- $y^2=8 x^6+17 x^5+32 x^4+15 x^3+36 x^2+19 x+5$
- $y^2=31 x^6+20 x^5+20 x^4+8 x^3+8 x^2+34 x+24$
- $y^2=17 x^6+8 x^5+15 x^4+29 x^3+x^2+26 x+36$
- $y^2=3 x^6+30 x^5+26 x^4+34 x^2+27 x+30$
- $y^2=23 x^6+12 x^5+18 x^4+18 x^3+22 x^2+33 x+29$
- $y^2=23 x^6+9 x^5+20 x^4+19 x^3+15 x^2+12 x+23$
- $y^2=17 x^6+11 x^5+6 x^4+32 x^3+32 x^2+21 x+15$
- $y^2=x^6+3 x^3+27$
- $y^2=15 x^6+28 x^5+13 x^4+34 x^3+22 x^2+4 x+2$
- $y^2=x^6+21 x^3+11$
- $y^2=x^6+24 x^5+4 x^4+16 x^3+28 x^2+29 x+10$
- $y^2=20 x^6+33 x^5+26 x^4+10 x^3+30 x^2+10 x+22$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{37}$.
Endomorphism algebra over $\F_{37}$| The isogeny class factors as 1.37.h 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$ |
Base change
This is a primitive isogeny class.