Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 12 x + 104 x^{2} + 444 x^{3} + 1369 x^{4}$ |
| Frobenius angles: | $\pm0.594270855772$, $\pm0.744393741071$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.5974272.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $24$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1930$ | $1964740$ | $2531297290$ | $3515666461200$ | $4808992908302650$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $50$ | $1434$ | $49970$ | $1875862$ | $69349850$ | $2565694986$ | $94931823050$ | $3512478427678$ | $129961763620850$ | $4808584260270714$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=5 x^6+20 x^5+33 x^4+29 x^3+10 x^2+29 x+12$
- $y^2=14 x^6+18 x^5+5 x^4+3 x^3+x^2+24 x+13$
- $y^2=21 x^6+19 x^5+22 x^4+16 x^3+18 x^2+31 x+25$
- $y^2=27 x^6+9 x^5+10 x^4+36 x^3+23 x^2+8 x+7$
- $y^2=7 x^6+16 x^5+15 x^4+24 x^3+29 x^2+22 x+27$
- $y^2=5 x^6+18 x^5+13 x^4+33 x^3+36 x^2+22 x+16$
- $y^2=31 x^6+15 x^5+17 x^4+4 x^3+x^2+x+3$
- $y^2=26 x^6+13 x^5+25 x^4+22 x^3+12 x^2+12 x+35$
- $y^2=28 x^6+32 x^5+34 x^4+26 x^3+35 x^2+33 x+25$
- $y^2=4 x^6+18 x^5+33 x^4+x^3+18 x^2+27 x+7$
- $y^2=27 x^6+20 x^5+15 x^4+15 x^3+27 x^2+5 x+5$
- $y^2=8 x^6+30 x^5+36 x^4+10 x^3+12 x^2+17 x+10$
- $y^2=21 x^6+18 x^5+17 x^4+20 x^3+28 x^2+20 x+17$
- $y^2=23 x^6+33 x^5+33 x^4+6 x^3+34 x^2+x+11$
- $y^2=12 x^6+22 x^5+23 x^4+35 x^3+31 x^2+7 x+9$
- $y^2=15 x^6+26 x^5+29 x^4+10 x^3+24 x^2+30 x+13$
- $y^2=36 x^6+6 x^5+29 x^4+21 x^2+27 x+26$
- $y^2=5 x^6+21 x^5+12 x^4+2 x^3+19 x^2+5 x+9$
- $y^2=30 x^6+4 x^5+31 x^4+5 x^3+30 x+21$
- $y^2=33 x^6+4 x^5+7 x^4+4 x^3+13 x^2+8 x+1$
- $y^2=32 x^6+32 x^5+23 x^4+4 x^3+27 x^2+29 x+11$
- $y^2=26 x^6+26 x^5+25 x^4+33 x^3+25 x^2+6 x+34$
- $y^2=14 x^6+34 x^5+29 x^4+26 x^3+17 x^2+10 x+32$
- $y^2=33 x^6+7 x^5+11 x^4+17 x^3+5 x^2+3 x+26$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{37}$.
Endomorphism algebra over $\F_{37}$| The endomorphism algebra of this simple isogeny class is 4.0.5974272.2. |
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.am_ea | $2$ | (not in LMFDB) |