Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 114 x^{2} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.0579565411254$, $\pm0.942043458875$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-2}, \sqrt{59})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $12$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3608$ | $13017664$ | $51520165400$ | $191553571922944$ | $713342912080733528$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $62$ | $3494$ | $226982$ | $13834734$ | $844596302$ | $51519956438$ | $3142742836022$ | $191707306686814$ | $11694146092834142$ | $713342912498584454$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=33 x^6+x^5+56 x^4+35 x^3+29 x^2+54 x+11$
- $y^2=26 x^6+7 x^5+5 x^4+6 x^3+27 x+28$
- $y^2=43 x^6+5 x^5+32 x^3+24 x^2+20 x+50$
- $y^2=31 x^6+30 x^5+6 x^4+50 x^3+46 x^2+17 x+55$
- $y^2=14 x^6+60 x^5+12 x^4+9 x^3+3 x^2+48 x+45$
- $y^2=16 x^6+26 x^5+42 x^4+27 x^3+30 x^2+17 x+43$
- $y^2=51 x^6+13 x^5+30 x^4+10 x^3+53 x^2+12 x+32$
- $y^2=28 x^5+47 x^4+12 x^3+7 x^2+7 x$
- $y^2=50 x^6+41 x^5+7 x^4+44 x^3+15 x^2+36 x+51$
- $y^2=29 x^6+37 x^5+3 x^4+44 x^3+17 x^2+2 x+49$
- $y^2=17 x^6+47 x^5+20 x^4+54 x^3+26 x^2+29 x+59$
- $y^2=52 x^6+26 x^5+28 x^4+36 x^3+21 x^2+36 x+21$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61^{2}}$.
Endomorphism algebra over $\F_{61}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{59})\). |
| The base change of $A$ to $\F_{61^{2}}$ is 1.3721.aek 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-118}) \)$)$ |
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.a_ek | $4$ | (not in LMFDB) |
| 2.61.ae_i | $8$ | (not in LMFDB) |
| 2.61.e_i | $8$ | (not in LMFDB) |