Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 12 x + 37 x^{2} )( 1 + 12 x + 37 x^{2} )$ |
| $1 - 70 x^{2} + 1369 x^{4}$ | |
| Frobenius angles: | $\pm0.0525684567113$, $\pm0.947431543289$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $11$ |
| Isomorphism classes: | 92 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1300$ | $1690000$ | $2565670900$ | $3504384000000$ | $4808584383596500$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $38$ | $1230$ | $50654$ | $1869838$ | $69343958$ | $2565615390$ | $94931877134$ | $3512477602078$ | $129961739795078$ | $4808584394775150$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):
- $y^2=12 x^6+4 x^5+2 x^4+17 x^2+7 x+25$
- $y^2=3 x^6+32 x^5+16 x^3+5 x+19$
- $y^2=25 x^6+23 x^5+20 x^4+32 x^2+24 x+21$
- $y^2=13 x^6+9 x^5+6 x^4+13 x^2+4 x+19$
- $y^2=23 x^6+27 x^5+29 x^4+23 x^3+10 x^2+26 x+10$
- $y^2=31 x^6+19 x^5+12 x^4+33 x^2+2 x+29$
- $y^2=5 x^6+31 x^5+24 x^4+14 x^2+19 x+19$
- $y^2=33 x^6+22 x^5+23 x^4+17 x^3+26 x^2+20 x+13$
- $y^2=6 x^6+17 x^5+10 x^4+2 x^3+5 x^2+32 x+10$
- $y^2=31 x^6+20 x^5+26 x^4+10 x^2+22 x+31$
- $y^2=21 x^6+17 x^5+25 x^4+22 x^3+17 x^2+2 x+15$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{37^{2}}$.
Endomorphism algebra over $\F_{37}$| The isogeny class factors as 1.37.am $\times$ 1.37.m and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
| The base change of $A$ to $\F_{37^{2}}$ is 1.1369.acs 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.