Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 26 x + 285 x^{2} - 1586 x^{3} + 3721 x^{4}$ |
Frobenius angles: | $\pm0.0471438380502$, $\pm0.263959254932$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.406080.2 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2395$ | $13457505$ | $51496571500$ | $191723273550345$ | $713333008278344875$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $36$ | $3616$ | $226878$ | $13846996$ | $844584576$ | $51519970078$ | $3142737883776$ | $191707276568356$ | $11694145959510198$ | $713342912228782576$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=52x^6+24x^5+15x^4+50x^3+10x^2+12x+6$
- $y^2=60x^6+38x^5+45x^4+36x^3+19x^2+48x+40$
- $y^2=25x^6+4x^5+4x^4+50x^3+31x^2+20x+31$
- $y^2=40x^6+4x^5+22x^4+55x^3+29x^2+9$
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 4.0.406080.2. |
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.ba_kz | $2$ | (not in LMFDB) |