Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x^{2} + 1369 x^{4}$ |
| Frobenius angles: | $\pm0.241392835167$, $\pm0.758607164833$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-70}, \sqrt{78})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $24$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1366$ | $1865956$ | $2565742774$ | $3522693549456$ | $4808584335371686$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $38$ | $1362$ | $50654$ | $1879606$ | $69343958$ | $2565759138$ | $94931877134$ | $3512472131998$ | $129961739795078$ | $4808584298325522$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=x^6+12 x^5+6 x^4+11 x^3+31 x^2+31 x+19$
- $y^2=2 x^6+24 x^5+12 x^4+22 x^3+25 x^2+25 x+1$
- $y^2=2 x^6+4 x^5+36 x^4+33 x^3+19 x^2+20 x+3$
- $y^2=4 x^6+8 x^5+35 x^4+29 x^3+x^2+3 x+6$
- $y^2=17 x^6+26 x^5+35 x^4+26 x^3+28 x^2+29 x+5$
- $y^2=34 x^6+15 x^5+33 x^4+15 x^3+19 x^2+21 x+10$
- $y^2=6 x^5+35 x^4+20 x^3+3 x^2+35 x+35$
- $y^2=12 x^5+33 x^4+3 x^3+6 x^2+33 x+33$
- $y^2=31 x^6+32 x^5+32 x^4+25 x^3+29 x^2+27 x+32$
- $y^2=25 x^6+27 x^5+27 x^4+13 x^3+21 x^2+17 x+27$
- $y^2=32 x^6+23 x^5+9 x^4+7 x^3+19 x^2+21 x+5$
- $y^2=27 x^6+9 x^5+18 x^4+14 x^3+x^2+5 x+10$
- $y^2=34 x^6+15 x^5+4 x^4+35 x^3+29 x^2+21 x+3$
- $y^2=31 x^6+30 x^5+8 x^4+33 x^3+21 x^2+5 x+6$
- $y^2=36 x^6+2 x^5+6 x^4+4 x^3+17 x^2+17 x+16$
- $y^2=35 x^6+4 x^5+12 x^4+8 x^3+34 x^2+34 x+32$
- $y^2=19 x^6+24 x^5+2 x^4+15 x^3+22 x^2+36 x+16$
- $y^2=x^6+11 x^5+4 x^4+30 x^3+7 x^2+35 x+32$
- $y^2=12 x^6+19 x^5+18 x^4+19 x^3+34 x^2+32 x+23$
- $y^2=24 x^6+x^5+36 x^4+x^3+31 x^2+27 x+9$
- $y^2=21 x^6+29 x^4+18 x^3+23 x^2+15 x$
- $y^2=5 x^6+21 x^4+36 x^3+9 x^2+30 x$
- $y^2=27 x^6+28 x^5+36 x^4+21 x^3+21 x^2+x+30$
- $y^2=17 x^6+19 x^5+35 x^4+5 x^3+5 x^2+2 x+23$
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 \(\Q(\sqrt{-70}, \sqrt{78})\). |
| The base change of $A$ to $\F_{37^{2}}$ is 1.1369.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1365}) \)$)$ |
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_e | $4$ | (not in LMFDB) |