Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 23 x + 239 x^{2} - 1403 x^{3} + 3721 x^{4}$ |
Frobenius angles: | $\pm0.0529194219962$, $\pm0.338378256587$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.4632645.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2535$ | $13656045$ | $51546134835$ | $191672000590005$ | $713281995600954000$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $39$ | $3671$ | $227097$ | $13843291$ | $844524174$ | $51519676223$ | $3142740010449$ | $191707323161011$ | $11694146290811127$ | $713342912420718326$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=42x^6+9x^5+54x^4+9x^3+54x^2+24x+35$
- $y^2=51x^6+16x^5+39x^4+53x^2+40x+35$
- $y^2=47x^6+12x^5+16x^4+26x^3+32x^2+14x+28$
- $y^2=24x^6+45x^5+49x^4+60x^3+29x^2+2x+55$
- $y^2=53x^6+57x^5+31x^4+2x^3+14x^2+13x+59$
- $y^2=21x^6+14x^5+27x^4+51x^3+8x^2+38x+11$
- $y^2=28x^6+13x^5+25x^4+51x^3+8x^2+16x+8$
- $y^2=24x^6+17x^5+9x^4+58x^3+49x^2+50x+37$
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 4.0.4632645.1. |
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.x_jf | $2$ | (not in LMFDB) |