Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 8 x + 80 x^{2} - 296 x^{3} + 1369 x^{4}$ |
| Frobenius angles: | $\pm0.299625924540$, $\pm0.478063709734$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.22790400.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1146$ | $2010084$ | $2592207306$ | $3511705191696$ | $4808201177155386$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $30$ | $1466$ | $51174$ | $1873750$ | $69338430$ | $2565737642$ | $94931575398$ | $3512475069214$ | $129961739748798$ | $4808584618133786$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=4 x^6+14 x^5+10 x^4+14 x^3+14 x^2+4 x+4$
- $y^2=13 x^6+9 x^5+17 x^4+10 x^3+17 x^2+30 x+33$
- $y^2=14 x^6+6 x^5+16 x^4+27 x^3+23 x^2+10 x+31$
- $y^2=7 x^6+8 x^5+25 x^4+25 x^3+4 x^2+14 x+23$
- $y^2=29 x^6+36 x^5+22 x^4+13 x^3+23 x^2+7 x+1$
- $y^2=36 x^6+32 x^5+25 x^4+23 x^3+10 x^2+27 x+33$
- $y^2=21 x^6+5 x^5+32 x^4+15 x^3+2 x^2+17 x+24$
- $y^2=19 x^6+4 x^5+6 x^4+15 x^3+9 x^2+16 x+22$
- $y^2=2 x^6+24 x^5+34 x^4+24 x^3+17 x^2+3 x+19$
- $y^2=24 x^6+9 x^5+32 x^4+35 x^3+36 x^2+20 x+27$
- $y^2=21 x^5+36 x^4+29 x^3+27 x^2+29 x+2$
- $y^2=10 x^6+18 x^5+16 x^4+4 x^3+31 x^2+30 x+1$
- $y^2=3 x^6+30 x^5+31 x^4+2 x^3+27 x^2+11 x+2$
- $y^2=23 x^6+31 x^5+17 x^4+16 x^3+27 x^2+8 x+1$
- $y^2=36 x^6+14 x^4+21 x^3+6 x^2+12 x+29$
- $y^2=3 x^6+x^5+29 x^4+28 x^3+36 x^2+33 x+30$
- $y^2=11 x^6+27 x^5+34 x^4+20 x^3+20 x^2+20 x+24$
- $y^2=29 x^6+29 x^5+13 x^4+14 x^3+9 x^2+3 x+13$
- $y^2=36 x^6+28 x^5+28 x^4+32 x^3+22 x^2+7 x+13$
- $y^2=6 x^6+18 x^5+25 x^4+17 x^3+13 x^2+3 x+26$
- $y^2=12 x^6+22 x^5+16 x^4+21 x^3+26 x^2+9 x+27$
- $y^2=30 x^5+11 x^4+24 x^3+30 x^2+34$
- $y^2=23 x^6+8 x^5+30 x^4+15 x^3+28 x^2+14 x+22$
- $y^2=27 x^6+29 x^5+8 x^4+2 x^3+10 x+14$
- $y^2=9 x^6+6 x^5+16 x^4+20 x^3+34 x^2+14 x+14$
- $y^2=x^6+3 x^5+9 x^4+23 x^3+18 x^2+9 x+32$
- $y^2=17 x^6+35 x^5+2 x^4+26 x^3+17 x^2+27 x+9$
- $y^2=5 x^6+9 x^5+24 x^4+x^3+29 x^2+8 x+17$
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 4.0.22790400.1. |
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.i_dc | $2$ | (not in LMFDB) |