Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 27 x + 303 x^{2} - 1647 x^{3} + 3721 x^{4}$ |
Frobenius angles: | $\pm0.114657586013$, $\pm0.208688212581$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.68725.1 |
Galois group: | $D_{4}$ |
Jacobians: | $5$ |
Isomorphism classes: | 5 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2351$ | $13398349$ | $51502277891$ | $191783445050389$ | $713402752074698576$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $35$ | $3599$ | $226901$ | $13851339$ | $844667150$ | $51520945943$ | $3142746156725$ | $191707325699539$ | $11694146107390571$ | $713342911549467254$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=41x^6+44x^5+17x^4+22x^3+60x^2+60x+8$
- $y^2=34x^6+32x^5+12x^4+42x^3+42x^2+36x+51$
- $y^2=42x^6+24x^5+38x^4+11x^3+55x^2+6x+54$
- $y^2=29x^6+46x^5+20x^4+12x^3+19x^2+20x+6$
- $y^2=43x^6+2x^5+54x^4+18x^3+32x^2+23x+48$
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 4.0.68725.1. |
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.bb_lr | $2$ | (not in LMFDB) |