Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 11 x + 100 x^{2} + 407 x^{3} + 1369 x^{4}$ |
| Frobenius angles: | $\pm0.591209716539$, $\pm0.713499096471$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.223397.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
| Isomorphism classes: | 28 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1888$ | $1986176$ | $2528077312$ | $3514967446016$ | $4809485300643808$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $49$ | $1449$ | $49906$ | $1875489$ | $69356949$ | $2565647094$ | $94931934433$ | $3512479651233$ | $129961746391690$ | $4808584365386929$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=12 x^6+10 x^5+25 x^3+35 x^2+22 x+5$
- $y^2=29 x^6+26 x^5+3 x^4+6 x^3+6 x^2+20 x$
- $y^2=31 x^6+10 x^5+33 x^4+10 x^3+27 x^2+24 x+20$
- $y^2=11 x^6+10 x^5+8 x^4+18 x^3+27 x^2+25 x+12$
- $y^2=35 x^6+32 x^5+25 x^4+14 x^3+26 x^2+24 x+32$
- $y^2=16 x^6+36 x^5+31 x^4+11 x^3+18 x^2+26 x+23$
- $y^2=34 x^6+33 x^5+22 x^4+9 x^3+35 x^2+36 x+16$
- $y^2=32 x^6+13 x^5+2 x^4+20 x^3+26 x^2+18 x+16$
- $y^2=24 x^6+13 x^5+32 x^4+x^3+3 x^2+33 x+22$
- $y^2=10 x^6+17 x^5+36 x^4+19 x^3+27 x^2+28 x+30$
- $y^2=3 x^5+15 x^4+x^3+10 x^2+22 x+1$
- $y^2=15 x^6+18 x^5+3 x^4+3 x^3+32 x^2+27 x+26$
- $y^2=22 x^6+22 x^5+16 x^4+21 x^3+5 x^2+13 x+9$
- $y^2=8 x^6+6 x^5+12 x^4+6 x^3+33 x+16$
- $y^2=23 x^6+10 x^5+12 x^4+28 x^3+5 x^2+22 x+13$
- $y^2=24 x^6+13 x^5+27 x^4+29 x^3+24 x^2+35 x+33$
- $y^2=4 x^6+7 x^5+4 x^4+22 x^3+10 x^2+1$
- $y^2=x^6+19 x^5+27 x^4+26 x^3+33 x^2+18 x+1$
- $y^2=10 x^6+21 x^5+32 x^4+31 x^3+34 x^2+30 x+24$
- $y^2=25 x^6+36 x^5+30 x^4+10 x^3+8 x^2+35 x+7$
- $y^2=28 x^6+16 x^5+13 x^4+14 x^3+18 x^2+18 x+30$
- $y^2=8 x^6+5 x^5+4 x^4+36 x^3+17 x^2+14 x+17$
- $y^2=33 x^6+10 x^5+19 x^4+8 x^3+29 x^2+30 x+9$
- $y^2=28 x^6+18 x^5+26 x^4+8 x^3+32 x^2+8 x+30$
- $y^2=26 x^6+11 x^5+26 x^4+28 x^3+18 x^2+21 x+4$
- $y^2=36 x^6+10 x^5+19 x^4+32 x^3+29 x^2+35 x+9$
- $y^2=8 x^6+x^5+16 x^4+31 x^3+33 x^2+8 x+9$
- $y^2=15 x^6+21 x^5+33 x^4+20 x^3+30 x^2+8 x+32$
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 4.0.223397.1. |
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.al_dw | $2$ | (not in LMFDB) |