Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 26 x + 289 x^{2} - 1586 x^{3} + 3721 x^{4}$ |
Frobenius angles: | $\pm0.125915170111$, $\pm0.234017939091$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.254528.3 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2399$ | $13489577$ | $51567838844$ | $191809495935353$ | $713404598749519959$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $36$ | $3624$ | $227190$ | $13853220$ | $844669336$ | $51520835550$ | $3142744647624$ | $191707315193988$ | $11694146086080174$ | $713342912009551624$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=10x^6+26x^5+20x^4+5x^3+11x^2+11x+32$
- $y^2=45x^6+52x^5+7x^4+60x^3+49x^2+9x+28$
- $y^2=23x^6+21x^5+35x^4+27x^3+32x^2+17x+54$
- $y^2=27x^6+41x^4+2x^3+55x^2+26x+9$
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 4.0.254528.3. |
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_ld | $2$ | (not in LMFDB) |