Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 15 x + 61 x^{2} )( 1 - 8 x + 61 x^{2} )$ |
$1 - 23 x + 242 x^{2} - 1403 x^{3} + 3721 x^{4}$ | |
Frobenius angles: | $\pm0.0900194921159$, $\pm0.328850104905$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $18$ |
Isomorphism classes: | 150 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2538$ | $13679820$ | $51593316768$ | $191719941336000$ | $713313755584881858$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $39$ | $3677$ | $227304$ | $13846753$ | $844561779$ | $51519979082$ | $3142742179119$ | $191707341720673$ | $11694146486324904$ | $713342914311311477$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=31x^6+19x^5+8x^4+43x^3+48x^2+10x+31$
- $y^2=13x^6+28x^5+13x^4+42x^3+32x^2+16x+4$
- $y^2=43x^6+27x^5+15x^4+49x^3+58x^2+56x+6$
- $y^2=32x^6+2x^5+36x^4+59x^3+41x^2+45x+31$
- $y^2=35x^6+5x^5+36x^4+46x^3+18x^2+37x+8$
- $y^2=46x^6+19x^5+28x^4+53x^3+27x^2+53x+47$
- $y^2=23x^6+8x^5+13x^4+32x^3+21x^2+35x$
- $y^2=21x^6+x^5+42x^4+19x^3+28x^2+15x+23$
- $y^2=40x^6+41x^5+58x^4+52x^3+49x^2+36x+44$
- $y^2=21x^6+5x^5+57x^4+53x^3+46x^2+48x+50$
- $y^2=8x^6+51x^5+4x^4+32x^3+42x^2+32x+26$
- $y^2=40x^6+22x^5+14x^4+x^3+58x^2+49x+10$
- $y^2=7x^6+23x^5+27x^4+24x^3+10x^2+40x+59$
- $y^2=30x^6+30x^5+45x^4+56x^3+22x^2+8x+20$
- $y^2=12x^6+57x^5+27x^4+27x^3+3x^2+9x+23$
- $y^2=43x^6+54x^5+34x^4+13x^3+18x^2+2x+50$
- $y^2=16x^6+60x^5+19x^4+43x^3+19x^2+31x+2$
- $y^2=11x^6+44x^5+14x^4+23x^3+34x^2+30x+56$
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.ai 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.ah_c | $2$ | (not in LMFDB) |
2.61.h_c | $2$ | (not in LMFDB) |
2.61.x_ji | $2$ | (not in LMFDB) |