Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 7 x + 82 x^{2} - 259 x^{3} + 1369 x^{4}$ |
| Frobenius angles: | $\pm0.348867809719$, $\pm0.462274902656$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.4937276.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $20$ |
| Isomorphism classes: | 20 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1186$ | $2037548$ | $2596391200$ | $3509570477504$ | $4807137618311146$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $31$ | $1485$ | $51256$ | $1872609$ | $69323091$ | $2565706170$ | $94932220183$ | $3512480236449$ | $129961740467512$ | $4808584429515925$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):
- $y^2=6 x^6+16 x^5+4 x^4+17 x^2+28 x+1$
- $y^2=30 x^6+9 x^5+2 x^3+24 x^2+28 x+24$
- $y^2=22 x^6+x^5+26 x^4+18 x^3+19 x^2+9 x+21$
- $y^2=29 x^6+11 x^5+6 x^4+30 x^3+15 x^2+25 x+8$
- $y^2=19 x^6+11 x^5+27 x^4+8 x^3+11 x^2+30 x+24$
- $y^2=33 x^6+28 x^5+17 x^4+8 x^3+20 x^2+21 x+29$
- $y^2=2 x^6+20 x^5+22 x^4+4 x^3+35 x^2+9 x+10$
- $y^2=5 x^6+14 x^5+21 x^4+34 x^3+2 x^2+9 x+2$
- $y^2=6 x^6+24 x^5+20 x^4+34 x^3+4 x^2+21 x+34$
- $y^2=18 x^6+12 x^5+8 x^4+16 x^3+35 x^2+12 x+17$
- $y^2=35 x^6+8 x^5+2 x^3+17 x^2+3 x+4$
- $y^2=31 x^6+5 x^5+23 x^4+19 x^3+17 x^2+20 x+24$
- $y^2=25 x^6+14 x^5+35 x^4+32 x^3+6 x^2+7 x+7$
- $y^2=24 x^6+34 x^5+11 x^4+x^3+17 x^2+26 x+17$
- $y^2=26 x^6+7 x^5+34 x^4+19 x^3+3 x^2+3 x+32$
- $y^2=19 x^6+8 x^5+19 x^3+25 x^2+11 x+13$
- $y^2=34 x^6+33 x^5+32 x^4+25 x^3+18 x^2+13 x+4$
- $y^2=36 x^6+31 x^5+27 x^4+4 x^3+x^2+23 x+8$
- $y^2=12 x^6+33 x^5+31 x^4+11 x^3+24 x^2+8 x+14$
- $y^2=10 x^6+x^5+2 x^4+2 x^3+7 x^2+20 x+10$
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 4.0.4937276.1. |
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 |
|---|---|---|
| 2.37.h_de | $2$ | (not in LMFDB) |