Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 14 x + 61 x^{2} )( 1 - 11 x + 61 x^{2} )$ |
| $1 - 25 x + 276 x^{2} - 1525 x^{3} + 3721 x^{4}$ | |
| Frobenius angles: | $\pm0.146275019398$, $\pm0.251304563322$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $10$ |
| Isomorphism classes: | 36 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2448$ | $13581504$ | $51634195200$ | $191837603153664$ | $713409532440053328$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $37$ | $3649$ | $227482$ | $13855249$ | $844675177$ | $51520784038$ | $3142743793597$ | $191707309147009$ | $11694146063548642$ | $713342911929317929$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=44x^6+52x^5+56x^4+57x^3+35x^2+34x+16$
- $y^2=7x^6+20x^5+56x^4+30x^3+35x^2+4x+21$
- $y^2=17x^6+17x^5+59x^4+3x^3+5x+19$
- $y^2=19x^6+24x^5+7x^4+30x^3+46x^2+3x+17$
- $y^2=26x^6+5x^5+10x^4+56x^3+9x^2+20x+24$
- $y^2=24x^6+60x^5+8x^4+50x^3+35x^2+26x+31$
- $y^2=52x^6+46x^5+33x^4+28x^3+13x^2+29x+11$
- $y^2=55x^6+9x^5+31x^4+2x^3+9x^2+6x+21$
- $y^2=5x^6+51x^5+22x^4+16x^3+3x^2+4$
- $y^2=13x^6+33x^5+54x^4+13x^3+7x^2+16x+23$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61}$.
Endomorphism algebra over $\F_{61}$| The isogeny class factors as 1.61.ao $\times$ 1.61.al 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.