Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 25 x + 270 x^{2} - 1525 x^{3} + 3721 x^{4}$ |
Frobenius angles: | $\pm0.0568212926978$, $\pm0.288610051981$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1267596.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2442$ | $13533564$ | $51531464952$ | $191720581827456$ | $713320847896463202$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $37$ | $3637$ | $227032$ | $13846801$ | $844570177$ | $51519854842$ | $3142738178197$ | $191707293412993$ | $11694146165782792$ | $713342913604307677$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+58x^5+15x^4+4x^3+52x^2+48x+24$
- $y^2=8x^6+46x^5+50x^4+39x^3+31x^2+29x+51$
- $y^2=7x^6+42x^5+58x^4+7x^3+35x^2+43x+42$
- $y^2=43x^6+58x^5+18x^4+18x^3+47x^2+47x+7$
- $y^2=38x^6+22x^5+5x^4+12x^3+13x^2+35x+49$
- $y^2=4x^6+20x^5+52x^4+3x^3+37x^2+31x+41$
- $y^2=5x^6+53x^5+13x^4+31x^3+34x^2+34x+48$
- $y^2=8x^6+3x^5+31x^4+59x^3+13x^2+8x+20$
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 4.0.1267596.1. |
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_kk | $2$ | (not in LMFDB) |