Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 24 x + 258 x^{2} - 1464 x^{3} + 3721 x^{4}$ |
Frobenius angles: | $\pm0.101801111500$, $\pm0.300249954030$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.15424.2 |
Galois group: | $D_{4}$ |
Jacobians: | $40$ |
Isomorphism classes: | 50 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2492$ | $13626256$ | $51602045852$ | $191760883557376$ | $713344109300063612$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $38$ | $3662$ | $227342$ | $13849710$ | $844597718$ | $51520160702$ | $3142741666046$ | $191707327722334$ | $11694146430022022$ | $713342915038512302$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 40 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=35x^6+60x^5+41x^4+24x^3+60x^2+5x+9$
- $y^2=40x^6+50x^5+60x^4+19x^3+32x^2+11x+17$
- $y^2=54x^6+35x^5+5x^4+x^3+50x^2+26x+38$
- $y^2=41x^6+21x^5+21x^4+36x^3+11x^2+19x+14$
- $y^2=31x^6+53x^5+22x^4+11x^3+47x^2+43$
- $y^2=52x^6+25x^5+14x^4+53x^3+43x^2+44x+34$
- $y^2=38x^6+12x^5+11x^4+34x^3+43x^2+38x+44$
- $y^2=11x^5+6x^4+9x^3+8x^2+30x+22$
- $y^2=18x^6+54x^5+52x^4+24x^3+57x^2+39x+29$
- $y^2=32x^6+25x^5+8x^4+3x^3+6x^2+36x+18$
- $y^2=19x^6+26x^5+21x^4+45x^3+23x^2+57x+56$
- $y^2=x^6+4x^5+24x^4+52x^3+53x^2+49x+38$
- $y^2=2x^6+51x^5+20x^4+53x^3+13x^2+32x+55$
- $y^2=35x^6+59x^5+52x^4+38x^3+9x^2+13x+44$
- $y^2=42x^6+24x^5+34x^4+39x^3+3x^2+x+23$
- $y^2=12x^6+33x^5+54x^4+2x^3+36x^2+48x+35$
- $y^2=47x^6+26x^5+35x^4+32x^3+46x^2+16x+59$
- $y^2=47x^6+27x^4+46x^3+57x^2+39x+55$
- $y^2=48x^6+14x^5+8x^4+14x^3+24x^2+59x+31$
- $y^2=22x^6+53x^5+60x^4+60x^3+34x^2+35x+58$
- and 20 more
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 4.0.15424.2. |
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_jy | $2$ | (not in LMFDB) |