Invariants
| Base field: | $\F_{71}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 23 x + 270 x^{2} + 1633 x^{3} + 5041 x^{4}$ |
| Frobenius angles: | $\pm0.689225310370$, $\pm0.797689990553$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.1409453.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $30$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6968$ | $25475008$ | $127541686688$ | $646150242812672$ | $3255116198266241608$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $95$ | $5053$ | $356348$ | $25427289$ | $1804158765$ | $128100192094$ | $9095123726819$ | $645753510559185$ | $45848500736549204$ | $3255243550814616693$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=22 x^6+64 x^5+42 x^4+54 x^3+35 x^2+36 x+65$
- $y^2=4 x^6+5 x^5+38 x^4+26 x^3+34 x^2+30 x+50$
- $y^2=62 x^6+58 x^5+51 x^4+62 x^3+49 x^2+30 x+4$
- $y^2=18 x^6+29 x^5+7 x^4+43 x^3+40 x^2+64 x+50$
- $y^2=69 x^6+21 x^5+22 x^4+64 x^3+25 x^2+8 x+58$
- $y^2=60 x^6+51 x^5+19 x^4+25 x^3+33 x^2+50 x+50$
- $y^2=14 x^6+16 x^5+43 x^4+37 x^3+54 x^2+32$
- $y^2=40 x^6+50 x^5+45 x^4+16 x^3+27 x^2+47 x+28$
- $y^2=19 x^6+3 x^5+50 x^4+40 x^3+17 x^2+60 x$
- $y^2=56 x^6+44 x^5+26 x^4+3 x^3+68 x^2+23 x+1$
- $y^2=18 x^6+43 x^5+57 x^4+13 x^3+35 x^2+61 x+15$
- $y^2=53 x^6+70 x^5+22 x^4+2 x^3+59 x^2+2 x+13$
- $y^2=30 x^6+6 x^5+12 x^4+x^3+15 x^2+13 x+24$
- $y^2=45 x^6+6 x^5+14 x^4+28 x^3+16 x^2+69 x+45$
- $y^2=25 x^6+70 x^5+59 x^4+6 x^3+25 x^2+62 x+22$
- $y^2=54 x^6+63 x^5+16 x^4+20 x^3+13 x^2+41 x+43$
- $y^2=43 x^6+23 x^5+8 x^4+29 x^3+35 x^2+16 x+25$
- $y^2=10 x^6+40 x^5+36 x^4+18 x^3+16 x^2+42 x+39$
- $y^2=60 x^6+20 x^5+43 x^4+40 x^3+15 x^2+34 x$
- $y^2=9 x^6+35 x^5+37 x^4+39 x^3+27 x^2+12 x+59$
- $y^2=6 x^6+13 x^5+39 x^4+11 x^3+34 x^2+20 x+40$
- $y^2=58 x^6+65 x^5+8 x^4+32 x^3+63 x^2+3 x+14$
- $y^2=22 x^6+29 x^5+63 x^4+3 x^3+32 x^2+63 x+44$
- $y^2=14 x^6+48 x^5+16 x^4+52 x^3+52 x^2+37 x+5$
- $y^2=21 x^6+53 x^5+41 x^4+11 x^3+51 x^2+15 x+22$
- $y^2=43 x^6+33 x^5+62 x^4+42 x^3+3 x^2+13 x+33$
- $y^2=31 x^6+63 x^5+63 x^4+26 x^3+16 x^2+54 x+11$
- $y^2=32 x^6+23 x^5+47 x^4+31 x^3+39 x^2+69 x+27$
- $y^2=52 x^6+36 x^5+52 x^4+66 x^3+59 x^2+52 x$
- $y^2=45 x^6+22 x^5+5 x^4+23 x^3+50 x^2+51 x+68$
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 4.0.1409453.1. |
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.ax_kk | $2$ | (not in LMFDB) |