Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 15 x + 61 x^{2} )( 1 - 9 x + 61 x^{2} )$ |
$1 - 24 x + 257 x^{2} - 1464 x^{3} + 3721 x^{4}$ | |
Frobenius angles: | $\pm0.0900194921159$, $\pm0.304548188780$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $11$ |
Isomorphism classes: | 20 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2491$ | $13618297$ | $51585620800$ | $191743238558025$ | $713331744448229731$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $38$ | $3660$ | $227270$ | $13848436$ | $844583078$ | $51520042662$ | $3142740999758$ | $191707325161636$ | $11694146420296190$ | $713342914941697980$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 11 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=4x^6+48x^5+39x^3+48x+4$
- $y^2=8x^6+36x^5+60x^4+9x^3+48x^2+59x+59$
- $y^2=4x^6+51x^5+36x^4+44x^3+33x^2+48x+38$
- $y^2=47x^6+3x^5+60x^4+44x^3+52x^2+15x+31$
- $y^2=42x^6+54x^5+51x^4+55x^3+50x^2+11x+21$
- $y^2=53x^6+10x^5+x^4+15x^3+34x^2+7x+31$
- $y^2=40x^6+53x^5+x^4+17x^3+46x^2+27x+30$
- $y^2=x^6+33x^5+12x^4+59x^3+6x^2+2x+26$
- $y^2=11x^6+38x^5+59x^4+27x^3+59x^2+38x+11$
- $y^2=21x^6+26x^5+52x^4+5x^3+52x^2+26x+21$
- $y^2=5x^6+6x^5+17x^4+10x^3+53x^2+39x+55$
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.aj 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.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.61.ag_an | $2$ | (not in LMFDB) |
2.61.g_an | $2$ | (not in LMFDB) |
2.61.y_jx | $2$ | (not in LMFDB) |