Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 56 x^{2} + 1369 x^{4}$ |
| Frobenius angles: | $\pm0.113391364009$, $\pm0.886608635991$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-2}, \sqrt{-65})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $24$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1314$ | $1726596$ | $2565780786$ | $3510991527696$ | $4808584499012514$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $38$ | $1258$ | $50654$ | $1873366$ | $69343958$ | $2565835162$ | $94931877134$ | $3512486633758$ | $129961739795078$ | $4808584625607178$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=15 x^6+13 x^5+27 x^4+3 x^3+20 x^2+16 x+20$
- $y^2=30 x^6+26 x^5+17 x^4+6 x^3+3 x^2+32 x+3$
- $y^2=36 x^6+11 x^5+23 x^4+6 x^3+24 x^2+11 x+6$
- $y^2=35 x^6+22 x^5+9 x^4+12 x^3+11 x^2+22 x+12$
- $y^2=24 x^6+3 x^5+24 x^4+2 x^3+6 x^2+8 x+22$
- $y^2=11 x^6+6 x^5+11 x^4+4 x^3+12 x^2+16 x+7$
- $y^2=17 x^6+12 x^5+25 x^4+7 x^3+31 x^2+x+8$
- $y^2=34 x^6+24 x^5+13 x^4+14 x^3+25 x^2+2 x+16$
- $y^2=4 x^6+x^5+30 x^4+16 x^3+32 x^2+30 x+24$
- $y^2=8 x^6+2 x^5+23 x^4+32 x^3+27 x^2+23 x+11$
- $y^2=6 x^6+17 x^5+16 x^4+11 x^3+3 x^2+15$
- $y^2=12 x^6+34 x^5+32 x^4+22 x^3+6 x^2+30$
- $y^2=12 x^6+8 x^5+11 x^4+5 x^3+25 x^2+13 x+13$
- $y^2=24 x^6+16 x^5+22 x^4+10 x^3+13 x^2+26 x+26$
- $y^2=11 x^6+30 x^5+31 x^4+6 x^3+13 x^2+5 x+31$
- $y^2=22 x^6+23 x^5+25 x^4+12 x^3+26 x^2+10 x+25$
- $y^2=13 x^6+x^5+34 x^4+19 x^3+5 x^2+36 x+23$
- $y^2=26 x^6+2 x^5+31 x^4+x^3+10 x^2+35 x+9$
- $y^2=18 x^6+3 x^5+28 x^4+30 x^3+17 x^2+22 x+9$
- $y^2=36 x^6+6 x^5+19 x^4+23 x^3+34 x^2+7 x+18$
- $y^2=31 x^6+11 x^5+33 x^4+22 x^3+26 x^2+17 x+28$
- $y^2=25 x^6+22 x^5+29 x^4+7 x^3+15 x^2+34 x+19$
- $y^2=28 x^6+20 x^5+9 x^4+9 x^3+28 x^2+9$
- $y^2=19 x^6+3 x^5+18 x^4+18 x^3+19 x^2+18$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{37^{2}}$.
Endomorphism algebra over $\F_{37}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{-65})\). |
| The base change of $A$ to $\F_{37^{2}}$ is 1.1369.ace 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-65}) \)$)$ |
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 |
|---|---|---|
| 2.37.a_ce | $4$ | (not in LMFDB) |
| 2.37.ag_s | $8$ | (not in LMFDB) |
| 2.37.g_s | $8$ | (not in LMFDB) |