Invariants
| Base field: | $\F_{71}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 26 x + 309 x^{2} - 1846 x^{3} + 5041 x^{4}$ |
| Frobenius angles: | $\pm0.173356710192$, $\pm0.258712538951$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.14912.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $9$ |
| Isomorphism classes: | 9 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3479$ | $25128817$ | $128454685604$ | $646154585275913$ | $3255476185108587559$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $46$ | $4984$ | $358900$ | $25427460$ | $1804358286$ | $128100877030$ | $9095119884274$ | $645753499323588$ | $45848500418096044$ | $3255243549568776024$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=52x^6+11x^5+55x^4+29x^3+9x^2+38x+7$
- $y^2=14x^6+57x^5+13x^4+2x^3+8x^2+70x+57$
- $y^2=17x^6+x^5+70x^4+10x^3+5x^2+42x+65$
- $y^2=46x^6+34x^5+34x^4+25x^3+32x^2+46x+46$
- $y^2=61x^6+61x^5+51x^4+69x^3+21x^2+5x+39$
- $y^2=7x^6+56x^5+16x^4+7x^3+10x+35$
- $y^2=23x^6+49x^5+67x^4+60x^3+58x^2+33x+8$
- $y^2=54x^6+29x^5+41x^4+54x^3+59x^2+60x+49$
- $y^2=39x^6+49x^5+22x^4+68x^3+57x^2+18x+28$
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 4.0.14912.2. |
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.ba_lx | $2$ | (not in LMFDB) |