Invariants
| Base field: | $\F_{67}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 27 x + 309 x^{2} - 1809 x^{3} + 4489 x^{4}$ |
| Frobenius angles: | $\pm0.0469987349196$, $\pm0.270485031029$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.737557.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $7$ |
| Isomorphism classes: | 7 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2963$ | $19659505$ | $90433720037$ | $406092338238925$ | $1822808788576916528$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $41$ | $4379$ | $300683$ | $20152347$ | $1350103616$ | $90457775111$ | $6060704435393$ | $406067627171683$ | $27206534249717591$ | $1822837805917807814$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):
- $y^2=38 x^6+66 x^5+64 x^4+8 x^3+51 x^2+47 x$
- $y^2=32 x^6+39 x^5+17 x^4+14 x^3+33 x^2+26 x+8$
- $y^2=50 x^6+38 x^5+24 x^4+24 x^3+41 x^2+x+1$
- $y^2=62 x^6+5 x^5+37 x^4+33 x^3+8 x^2+24 x+18$
- $y^2=37 x^6+63 x^5+18 x^4+21 x^3+53 x^2+41 x+28$
- $y^2=41 x^6+13 x^5+63 x^4+26 x^3+66 x^2+21 x+28$
- $y^2=53 x^6+44 x^5+65 x^4+59 x^3+25 x^2+48 x+20$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{67}$.
Endomorphism algebra over $\F_{67}$| The endomorphism algebra of this simple isogeny class is 4.0.737557.1. |
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 |
|---|---|---|
| 2.67.bb_lx | $2$ | (not in LMFDB) |