Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 25 x + 277 x^{2} - 1525 x^{3} + 3721 x^{4}$ |
Frobenius angles: | $\pm0.162949397008$, $\pm0.240145470170$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.167525.1 |
Galois group: | $D_{4}$ |
Jacobians: | $9$ |
Isomorphism classes: | 9 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2449$ | $13589501$ | $51651322669$ | $191856914855525$ | $713423574578330704$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $37$ | $3651$ | $227557$ | $13856643$ | $844691802$ | $51520911051$ | $3142744131697$ | $191707302173283$ | $11694145926427417$ | $713342910479846806$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=57x^6+29x^5+12x^4+55x^3+13x^2+4x+30$
- $y^2=15x^6+7x^5+22x^4+10x^3+30x^2+47x+34$
- $y^2=35x^6+10x^5+33x^4+19x^3+51x^2+41x+50$
- $y^2=7x^6+60x^5+x^4+3x^3+45x^2+59x+45$
- $y^2=33x^6+44x^5+45x^4+3x^3+15x^2+22x+51$
- $y^2=12x^6+22x^5+x^4+32x^3+47x^2+49x+50$
- $y^2=7x^6+60x^5+7x^4+8x^3+44x^2+18x+18$
- $y^2=17x^6+8x^5+20x^4+60x^3+22x^2+33x+28$
- $y^2=50x^6+29x^5+25x^4+18x^3+45x^2+8x+43$
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 4.0.167525.1. |
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.z_kr | $2$ | (not in LMFDB) |