Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 16 x + 123 x^{2} - 496 x^{3} + 961 x^{4}$ |
| Frobenius angles: | $\pm0.161541127494$, $\pm0.309693907298$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.357264.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $573$ | $915081$ | $897091092$ | $854979395001$ | $819834001771173$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $16$ | $952$ | $30112$ | $925780$ | $28636336$ | $887509222$ | $27512631568$ | $852892028644$ | $26439631716832$ | $819628321295752$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=29 x^6+28 x^5+28 x^4+13 x^3+6 x^2+14 x+26$
- $y^2=12 x^6+29 x^5+8 x^4+15 x^3+10 x^2+22 x+11$
- $y^2=21 x^6+14 x^5+11 x^4+28 x^3+18 x^2+17 x+12$
- $y^2=29 x^6+21 x^5+5 x^4+7 x^3+26 x^2+3 x+19$
- $y^2=15 x^6+27 x^5+25 x^4+2 x^3+13 x^2+28 x+25$
- $y^2=11 x^6+16 x^5+26 x^4+19 x^3+3 x^2+18 x+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{31}$.
Endomorphism algebra over $\F_{31}$| The endomorphism algebra of this simple isogeny class is 4.0.357264.1. |
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 |
|---|---|---|
| 2.31.q_et | $2$ | (not in LMFDB) |