Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 5 x + 37 x^{2} )^{2}$ |
| $1 + 10 x + 99 x^{2} + 370 x^{3} + 1369 x^{4}$ | |
| Frobenius angles: | $\pm0.634819497847$, $\pm0.634819497847$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $12$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1849$ | $2013561$ | $2522450176$ | $3513746501001$ | $4810557549061489$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $48$ | $1468$ | $49794$ | $1874836$ | $69372408$ | $2565559222$ | $94931660424$ | $3512486723428$ | $129961711465818$ | $4808584245092428$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=28 x^6+17 x^5+4 x^4+24 x^3+33 x^2+17 x+9$
- $y^2=4 x^6+3 x^4+34 x^3+2 x^2+34 x+1$
- $y^2=5 x^6+8 x^4+17 x^3+7 x^2+26 x+14$
- $y^2=25 x^6+36 x^5+22 x^4+16 x^3+25 x+8$
- $y^2=3 x^6+34 x^5+24 x^4+18 x^3+35 x^2+10 x+7$
- $y^2=11 x^6+12 x^5+18 x^4+2 x^3+30 x^2+36 x+4$
- $y^2=4 x^6+34 x^5+19 x^4+12 x^3+2 x^2+17 x+9$
- $y^2=2 x^6+11 x^5+9 x^4+31 x^3+28 x^2+5 x+9$
- $y^2=30 x^6+32 x^5+16 x^4+20 x^3+12 x^2+x+21$
- $y^2=4 x^6+36 x^5+x^4+7 x^3+11 x^2+27 x+33$
- $y^2=6 x^6+16 x^5+20 x^4+7 x^3+32 x^2+x+8$
- $y^2=2 x^6+14 x^3+20$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{37}$.
Endomorphism algebra over $\F_{37}$| The isogeny class factors as 1.37.f 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$ |
Base change
This is a primitive isogeny class.