Invariants
| Base field: | $\F_{71}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 84 x^{2} + 5041 x^{4}$ |
| Frobenius angles: | $\pm0.149258255063$, $\pm0.850741744937$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-58}, \sqrt{226})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $14$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4958$ | $24581764$ | $128100961550$ | $645907382725264$ | $3255243551093960078$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $72$ | $4874$ | $357912$ | $25417734$ | $1804229352$ | $128101639178$ | $9095120158392$ | $645753614579134$ | $45848500718449032$ | $3255243551178038954$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):
- $y^2=52 x^6+31 x^5+43 x^4+45 x^3+16 x^2+29 x+64$
- $y^2=9 x^6+4 x^5+17 x^4+31 x^3+41 x^2+61 x+22$
- $y^2=44 x^6+23 x^5+45 x^4+x^3+38 x^2+32 x+20$
- $y^2=24 x^6+19 x^5+31 x^4+7 x^3+53 x^2+11 x+69$
- $y^2=56 x^6+55 x^5+55 x^4+2 x^3+64 x^2+28 x+23$
- $y^2=37 x^6+30 x^5+30 x^4+14 x^3+22 x^2+54 x+19$
- $y^2=47 x^6+47 x^5+50 x^4+9 x^3+4 x^2+66 x+49$
- $y^2=45 x^6+45 x^5+66 x^4+63 x^3+28 x^2+36 x+59$
- $y^2=55 x^6+66 x^5+12 x^4+31 x^3+30 x^2+57 x+22$
- $y^2=30 x^6+36 x^5+13 x^4+4 x^3+68 x^2+44 x+12$
- $y^2=68 x^6+4 x^5+66 x^4+57 x^3+3 x^2+34 x+36$
- $y^2=50 x^6+28 x^5+36 x^4+44 x^3+21 x^2+25 x+39$
- $y^2=55 x^6+39 x^5+15 x^4+58 x^3+22 x^2+45$
- $y^2=30 x^6+60 x^5+34 x^4+51 x^3+12 x^2+31$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{71^{2}}$.
Endomorphism algebra over $\F_{71}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-58}, \sqrt{226})\). |
| The base change of $A$ to $\F_{71^{2}}$ is 1.5041.adg 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3277}) \)$)$ |
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 |
|---|---|---|
| 2.71.a_dg | $4$ | (not in LMFDB) |