Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 25 x + 275 x^{2} - 1525 x^{3} + 3721 x^{4}$ |
Frobenius angles: | $\pm0.131683462108$, $\pm0.259880985593$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.863421.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2447$ | $13573509$ | $51617069387$ | $191818236647781$ | $713395279273222352$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $37$ | $3647$ | $227407$ | $13853851$ | $844658302$ | $51520649087$ | $3142743286447$ | $191707313457043$ | $11694146168714917$ | $713342913098114102$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=23x^6+56x^5+14x^4+24x^3+47x^2+12x+26$
- $y^2=8x^6+48x^5+39x^4+42x^3+2x^2+22x$
- $y^2=23x^6+39x^5+8x^4+51x^3+19x^2+55x+11$
- $y^2=38x^6+3x^5+15x^4+52x^3+3x^2+10x+31$
- $y^2=41x^6+9x^5+25x^4+40x^3+14x^2+19x+8$
- $y^2=47x^6+2x^5+28x^3+37x^2+27x+35$
- $y^2=40x^6+29x^5+2x^4+29x^3+50x^2+47x+24$
- $y^2=33x^6+23x^5+47x^4+26x^3+48x^2+14x+51$
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 4.0.863421.1. |
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_kp | $2$ | (not in LMFDB) |