Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 12 x + 37 x^{2} )( 1 - 5 x + 37 x^{2} )$ |
| $1 - 17 x + 134 x^{2} - 629 x^{3} + 1369 x^{4}$ | |
| Frobenius angles: | $\pm0.0525684567113$, $\pm0.365180502153$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $4$ |
| Isomorphism classes: | 26 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $858$ | $1844700$ | $2567379672$ | $3509062128000$ | $4806815221958178$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $21$ | $1349$ | $50688$ | $1872337$ | $69318441$ | $2565587306$ | $94931737173$ | $3512482162753$ | $129961752036336$ | $4808584319933789$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=17x^6+3x^5+31x^4+22x^3+13x^2+20x+13$
- $y^2=3x^6+22x^5+9x^4+18x^3+3x^2+24x+32$
- $y^2=12x^6+19x^5+2x^4+13x^3+34x^2+12x+13$
- $y^2=35x^6+36x^5+8x^4+36x^3+26x^2+18x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{37}$.
Endomorphism algebra over $\F_{37}$| The isogeny class factors as 1.37.am $\times$ 1.37.af 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.