Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 12 x + 72 x^{2} + 372 x^{3} + 961 x^{4}$ |
| Frobenius angles: | $\pm0.525782775241$, $\pm0.974217224759$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{26})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $9$ |
| Isomorphism classes: | 18 |
| Cyclic group of points: | yes |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1418$ | $921700$ | $895043018$ | $849530890000$ | $820030682554058$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $44$ | $962$ | $30044$ | $919878$ | $28643204$ | $887503682$ | $27512715764$ | $852888092158$ | $26439636681164$ | $819628286980802$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=2 x^6+24 x^5+9 x^4+16 x^3+4 x^2+7 x+16$
- $y^2=9 x^6+20 x^5+10 x^4+10 x^2+11 x+9$
- $y^2=9 x^6+2 x^5+30 x^4+29 x^3+17 x^2+x+9$
- $y^2=17 x^6+6 x^5+8 x^4+15 x^3+9 x^2+14 x+3$
- $y^2=24 x^6+11 x^5+15 x^4+26 x^3+29 x+29$
- $y^2=10 x^6+29 x^5+20 x^4+20 x^2+2 x+10$
- $y^2=10 x^6+6 x^5+2 x^4+30 x^3+6 x^2+10 x+17$
- $y^2=24 x^6+7 x^5+28 x^4+28 x^2+24 x+24$
- $y^2=10 x^6+7 x^5+12 x^4+5 x^3+8 x^2+7$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{31^{4}}$.
Endomorphism algebra over $\F_{31}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{26})\). |
| The base change of $A$ to $\F_{31^{4}}$ is 1.923521.acsc 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-26}) \)$)$ |
- Endomorphism algebra over $\F_{31^{2}}$
The base change of $A$ to $\F_{31^{2}}$ is the simple isogeny class 2.961.a_acsc and its endomorphism algebra is \(\Q(i, \sqrt{26})\).
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.am_cu | $2$ | (not in LMFDB) |
| 2.31.a_ak | $8$ | (not in LMFDB) |
| 2.31.a_k | $8$ | (not in LMFDB) |