Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 13 x + 108 x^{2} + 481 x^{3} + 1369 x^{4}$ |
| Frobenius angles: | $\pm0.596385117745$, $\pm0.779941848074$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.2208492.2 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1972$ | $1940448$ | $2536930672$ | $3515447547264$ | $4808639237567812$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $51$ | $1417$ | $50082$ | $1875745$ | $69344751$ | $2565755926$ | $94931429427$ | $3512479537825$ | $129961771773882$ | $4808584122571537$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=9 x^6+2 x^5+15 x^4+13 x^3+x^2+23 x+2$
- $y^2=21 x^6+22 x^5+24 x^4+28 x^3+17 x^2+23 x+4$
- $y^2=20 x^5+16 x^4+28 x^3+30 x^2+7 x$
- $y^2=21 x^6+17 x^5+6 x^4+9 x^3+9 x^2+12 x+30$
- $y^2=22 x^6+12 x^5+14 x^4+34 x^3+28 x^2+18 x+27$
- $y^2=25 x^6+4 x^5+15 x^4+6 x^3+25 x^2+24 x+30$
- $y^2=24 x^6+6 x^5+32 x^4+15 x^3+7 x^2+11 x+23$
- $y^2=x^6+30 x^5+25 x^4+26 x^3+15 x^2+18 x+21$
- $y^2=5 x^6+9 x^4+31 x^3+18 x^2+34 x+26$
- $y^2=5 x^6+24 x^5+13 x^4+17 x^3+11 x^2+22 x+6$
- $y^2=30 x^6+3 x^5+16 x^4+17 x^3+20 x^2+17 x+3$
- $y^2=34 x^6+13 x^5+17 x^4+29 x^3+32 x^2+26 x+14$
- $y^2=15 x^6+10 x^5+13 x^4+32 x^3+30 x^2+16 x+30$
- $y^2=12 x^6+34 x^5+17 x^4+10 x^3+13 x^2+x+26$
- $y^2=15 x^5+34 x^4+17 x^3+35 x^2+6 x+23$
- $y^2=23 x^6+23 x^5+32 x^4+9 x^3+25 x^2+10 x+11$
- $y^2=24 x^6+3 x^5+26 x^4+16 x^3+4 x^2+31 x+6$
- $y^2=33 x^6+22 x^5+11 x^4+34 x^3+10 x^2+27 x+4$
- $y^2=9 x^6+27 x^5+27 x^4+12 x^3+20 x^2+5 x+21$
- $y^2=30 x^6+12 x^5+28 x^4+12 x^3+14 x^2+14 x+12$
- $y^2=12 x^6+26 x^5+10 x^4+12 x^3+25 x+25$
- $y^2=29 x^6+13 x^5+16 x^4+35 x^3+x^2+12 x+21$
- $y^2=5 x^6+24 x^5+2 x^4+29 x^3+25 x^2+x+6$
- $y^2=19 x^6+26 x^5+16 x^4+27 x^3+6 x^2+14 x+17$
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 4.0.2208492.2. |
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.an_ee | $2$ | (not in LMFDB) |