Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 24 x + 254 x^{2} - 1464 x^{3} + 3721 x^{4}$ |
Frobenius angles: | $\pm0.0450812475667$, $\pm0.315976986299$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.29952.1 |
Galois group: | $D_{4}$ |
Jacobians: | $14$ |
Isomorphism classes: | 22 |
This isogeny class is simple and geometrically 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2488$ | $13594432$ | $51536354872$ | $191689974125568$ | $713293434159593848$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $38$ | $3654$ | $227054$ | $13844590$ | $844537718$ | $51519644790$ | $3142738105790$ | $191707303674718$ | $11694146220469190$ | $713342912883732774$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=43x^6+31x^5+5x^4+9x^3+15x^2+6x+20$
- $y^2=32x^6+46x^5+18x^4+40x^3+26x^2+11x+10$
- $y^2=35x^6+15x^5+23x^4+35x^3+48x^2+39x+23$
- $y^2=40x^6+40x^5+39x^4+57x^3+30x^2+10x+10$
- $y^2=35x^6+42x^5+27x^4+49x^3+59x^2+9x+29$
- $y^2=28x^6+12x^5+17x^4+26x^3+50x^2+56x$
- $y^2=18x^6+x^5+40x^4+15x^3+14x^2+34x+11$
- $y^2=47x^6+58x^5+36x^4+22x^3+9x^2+36x+45$
- $y^2=50x^6+50x^5+55x^4+36x^3+57x^2+34x+12$
- $y^2=50x^6+27x^5+46x^4+32x^3+26x^2+25x+44$
- $y^2=8x^6+35x^5+45x^4+42x^3+33x^2+6x+50$
- $y^2=34x^6+48x^5+57x^4+24x^3+43x^2+44x+25$
- $y^2=7x^6+39x^5+20x^4+19x^3+42x^2+4x+26$
- $y^2=44x^6+54x^5+43x^4+23x^3+7x^2+15x+29$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61}$.
Endomorphism algebra over $\F_{61}$The endomorphism algebra of this simple isogeny class is 4.0.29952.1. |
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.y_ju | $2$ | (not in LMFDB) |