Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 15 x + 61 x^{2} )( 1 - 12 x + 61 x^{2} )$ |
$1 - 27 x + 302 x^{2} - 1647 x^{3} + 3721 x^{4}$ | |
Frobenius angles: | $\pm0.0900194921159$, $\pm0.221142061624$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $5$ |
Isomorphism classes: | 13 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2350$ | $13390300$ | $51483762400$ | $191759808240000$ | $713381657019583750$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $35$ | $3597$ | $226820$ | $13849633$ | $844642175$ | $51520666362$ | $3142743712955$ | $191707309936993$ | $11694146055603140$ | $713342911939629477$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=16x^6+22x^5+3x^4+43x^3+21x^2+50x+43$
- $y^2=29x^6+21x^5+35x^4+47x^3+12x^2+12x+37$
- $y^2=38x^6+43x^5+3x^4+2x^3+8x^2+36x+56$
- $y^2=43x^6+34x^5+21x^4+49x^2+34x+24$
- $y^2=21x^6+37x^5+21x^4+57x^3+9x^2+41x+41$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61}$.
Endomorphism algebra over $\F_{61}$The isogeny class factors as 1.61.ap $\times$ 1.61.am 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.