Invariants
| Base field: | $\F_{71}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 20 x + 236 x^{2} + 1420 x^{3} + 5041 x^{4}$ |
| Frobenius angles: | $\pm0.647877913640$, $\pm0.764579467144$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-178 +20 \sqrt{6}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $20$ |
| Isomorphism classes: | 20 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6718$ | $25783684$ | $127421817550$ | $646078168411024$ | $3255225327426443278$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $92$ | $5114$ | $356012$ | $25424454$ | $1804219252$ | $128099806778$ | $9095123526532$ | $645753526447294$ | $45848500747995932$ | $3255243549173905754$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):
- $y^2=31 x^6+69 x^5+46 x^4+56 x^3+27 x^2+55 x+36$
- $y^2=4 x^6+41 x^5+3 x^4+2 x^3+45 x^2+12 x+64$
- $y^2=9 x^6+50 x^5+29 x^4+15 x^3+16 x^2+69 x+18$
- $y^2=32 x^6+47 x^5+29 x^4+64 x^3+35 x^2+10 x+27$
- $y^2=43 x^6+15 x^5+x^4+58 x^3+69 x^2+22 x+30$
- $y^2=60 x^6+50 x^5+47 x^4+22 x^3+18 x^2+10 x+62$
- $y^2=12 x^6+31 x^5+44 x^4+34 x^3+25 x^2+25 x+29$
- $y^2=44 x^6+9 x^5+11 x^4+66 x^3+43 x^2+33 x+49$
- $y^2=56 x^6+51 x^4+27 x^3+45 x^2+66 x+51$
- $y^2=8 x^6+37 x^5+16 x^4+17 x^3+33 x^2+36 x+67$
- $y^2=8 x^6+43 x^5+38 x^4+61 x^3+61 x^2+7 x+49$
- $y^2=55 x^6+58 x^5+6 x^4+4 x^3+12 x^2+37 x+65$
- $y^2=54 x^6+13 x^5+19 x^4+14 x^3+10 x^2+27 x+7$
- $y^2=60 x^6+61 x^5+41 x^4+24 x^3+62 x^2+4 x+27$
- $y^2=20 x^6+11 x^5+35 x^4+14 x^3+70 x^2+45 x+4$
- $y^2=17 x^6+62 x^5+53 x^4+21 x^3+61 x^2+5 x$
- $y^2=68 x^6+39 x^5+33 x^4+69 x^3+70 x^2+69 x+51$
- $y^2=40 x^6+54 x^5+27 x^4+31 x^3+10 x^2+53 x+22$
- $y^2=57 x^6+20 x^5+56 x^4+55 x^3+13 x^2+38 x+66$
- $y^2=47 x^6+64 x^5+62 x^4+23 x^3+42 x^2+46 x+40$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{71}$.
Endomorphism algebra over $\F_{71}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-178 +20 \sqrt{6}})\). |
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.au_jc | $2$ | (not in LMFDB) |