Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 10 x + 96 x^{2} - 370 x^{3} + 1369 x^{4}$ |
| Frobenius angles: | $\pm0.313341122326$, $\pm0.413431169513$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-120 -10 \sqrt{3}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $16$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1086$ | $2004756$ | $2604976254$ | $3513687727056$ | $4807215024358686$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $28$ | $1462$ | $51424$ | $1874806$ | $69324208$ | $2565626038$ | $94931953564$ | $3512481664798$ | $129961744466668$ | $4808584372578022$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):
- $y^2=23 x^6+27 x^5+21 x^4+5 x^3+2 x^2+35 x+8$
- $y^2=14 x^6+x^5+27 x^4+9 x^3+9 x^2+26 x+15$
- $y^2=20 x^6+31 x^5+17 x^4+35 x^3+7 x^2+19$
- $y^2=5 x^6+x^5+30 x^4+19 x^3+9 x^2+24 x+28$
- $y^2=5 x^6+36 x^5+2 x^4+4 x^3+8 x^2+12 x+5$
- $y^2=24 x^6+14 x^5+33 x^4+20 x^3+29 x^2+33 x+31$
- $y^2=14 x^6+4 x^5+36 x^4+19 x^3+26 x^2+7 x+13$
- $y^2=16 x^6+6 x^5+29 x^4+18 x^3+7 x^2+28 x+17$
- $y^2=35 x^6+29 x^5+11 x^4+33 x^3+12 x^2+18 x+32$
- $y^2=17 x^6+22 x^5+30 x^4+5 x^3+34 x^2+12 x+2$
- $y^2=13 x^6+7 x^5+29 x^4+13 x^3+5 x+14$
- $y^2=6 x^6+33 x^5+23 x^4+13 x^3+4 x^2+33 x+27$
- $y^2=19 x^6+14 x^5+23 x^4+8 x^3+31 x^2+12 x+27$
- $y^2=18 x^6+15 x^5+10 x^4+16 x^3+16 x^2+3 x+35$
- $y^2=28 x^6+22 x^5+5 x^4+31 x^3+3 x^2+12 x+4$
- $y^2=24 x^6+24 x^5+24 x^4+9 x^3+25 x^2+6 x+7$
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 \(\Q(\sqrt{-120 -10 \sqrt{3}})\). |
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.k_ds | $2$ | (not in LMFDB) |