Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 24 x + 264 x^{2} - 1464 x^{3} + 3721 x^{4}$ |
Frobenius angles: | $\pm0.171237183552$, $\pm0.262983076991$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.540928.1 |
Galois group: | $D_{4}$ |
Jacobians: | $16$ |
Isomorphism classes: | 16 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2498$ | $13674052$ | $51700628882$ | $191865600966928$ | $713414043543619298$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $38$ | $3674$ | $227774$ | $13857270$ | $844680518$ | $51520716650$ | $3142742601470$ | $191707297772638$ | $11694145977106310$ | $713342911287682874$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=20x^6+24x^5+28x^4+12x^3+51x^2+8x+8$
- $y^2=44x^6+29x^5+45x^4+13x^3+45x^2+10x+33$
- $y^2=31x^5+56x^4+27x^3+31x^2+2x+8$
- $y^2=28x^6+21x^5+9x^4+48x^3+34x^2+46x+58$
- $y^2=59x^6+27x^5+42x^4+35x^3+44x^2+52x+22$
- $y^2=11x^6+13x^5+39x^4+45x^3+28x^2+21x+16$
- $y^2=50x^6+27x^5+53x^4+58x^3+10x^2+2x$
- $y^2=49x^6+4x^5+41x^4+9x^3+2x^2+48x+2$
- $y^2=10x^6+38x^5+30x^4+44x^3+36x^2+58x+10$
- $y^2=23x^6+50x^5+41x^4+39x^3+55x^2+8x+31$
- $y^2=8x^6+54x^5+28x^4+9x^3+35x^2+34x+44$
- $y^2=23x^6+9x^5+21x^4+5x^2+8$
- $y^2=54x^6+31x^5+58x^4+6x^3+27x^2+42x+24$
- $y^2=50x^6+47x^5+21x^4+36x^3+17x^2+14x+53$
- $y^2=8x^6+2x^5+33x^4+2x^3+12x^2+13x+51$
- $y^2=7x^6+54x^5+43x^4+54x^3+30x^2+18x+27$
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 4.0.540928.1. |
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_ke | $2$ | (not in LMFDB) |