Invariants
| Base field: | $\F_{71}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 20 x + 239 x^{2} + 1420 x^{3} + 5041 x^{4}$ |
| Frobenius angles: | $\pm0.663226938123$, $\pm0.745113642058$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.4544784.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6721$ | $25815361$ | $127357600084$ | $646127739735049$ | $3255227494412466001$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $92$ | $5120$ | $355832$ | $25426404$ | $1804220452$ | $128099503766$ | $9095127468652$ | $645753509922244$ | $45848500550453672$ | $3255243553018306400$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):
- $y^2=16 x^6+16 x^5+9 x^4+58 x^3+23 x^2+26 x+48$
- $y^2=44 x^6+31 x^5+17 x^4+23 x^3+18 x^2+56 x+49$
- $y^2=67 x^6+56 x^5+40 x^4+50 x^3+32 x^2+30 x+43$
- $y^2=30 x^6+10 x^5+50 x^4+70 x^3+8 x^2+4 x+6$
- $y^2=69 x^6+8 x^5+67 x^4+62 x^3+5 x^2+63 x+64$
- $y^2=68 x^6+60 x^5+2 x^4+51 x^3+32 x^2+3 x+43$
- $y^2=54 x^6+37 x^5+31 x^4+x^3+68 x^2+44 x+5$
- $y^2=23 x^6+2 x^5+51 x^4+35 x^3+41 x^2+41 x+39$
- $y^2=34 x^6+50 x^5+29 x^4+33 x^3+13 x^2+10 x+53$
- $y^2=10 x^6+9 x^5+30 x^4+23 x^3+32 x^2+65 x+30$
- $y^2=12 x^6+34 x^5+3 x^4+36 x^3+x^2+35 x+70$
- $y^2=48 x^6+30 x^5+61 x^4+43 x^3+24 x^2+14 x+12$
- $y^2=48 x^6+61 x^5+63 x^4+54 x^3+63 x^2+62 x+65$
- $y^2=28 x^6+45 x^5+11 x^3+41 x^2+46 x+2$
- $y^2=35 x^6+18 x^4+38 x^3+30 x^2+69 x+52$
- $y^2=45 x^6+52 x^5+16 x^4+56 x^3+7 x^2+40 x+15$
- $y^2=8 x^6+9 x^5+16 x^4+67 x^3+66 x^2+55 x+8$
- $y^2=29 x^6+40 x^5+31 x^4+51 x^3+8 x^2+41 x+17$
- $y^2=46 x^6+42 x^5+x^4+58 x^3+32 x^2+64 x+27$
- $y^2=31 x^6+13 x^5+28 x^4+5 x^3+11 x^2+35 x+26$
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 4.0.4544784.1. |
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_jf | $2$ | (not in LMFDB) |