Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 12 x + 37 x^{2} )( 1 - 6 x + 37 x^{2} )$ |
| $1 - 18 x + 146 x^{2} - 666 x^{3} + 1369 x^{4}$ | |
| Frobenius angles: | $\pm0.0525684567113$, $\pm0.335828188403$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $10$ |
| Isomorphism classes: | 46 |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $832$ | $1830400$ | $2568384832$ | $3510853632000$ | $4807185389061952$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $1338$ | $50708$ | $1873294$ | $69323780$ | $2565569706$ | $94931350436$ | $3512480601886$ | $129961760615156$ | $4808584443323418$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=5 x^6+25 x^4+25 x^3+19 x^2+18 x+18$
- $y^2=8 x^6+25 x^5+18 x^4+28 x^3+15 x^2+33 x+23$
- $y^2=x^5+6 x^4+35 x^3+34 x^2+19 x+20$
- $y^2=29 x^6+33 x^5+17 x^4+14 x^3+3 x^2+3 x+18$
- $y^2=5 x^6+29 x^5+4 x^4+27 x^3+26 x^2+24 x+19$
- $y^2=32 x^6+27 x^5+23 x^4+31 x^3+10 x^2+24 x+30$
- $y^2=2 x^6+25 x^5+29 x^4+29 x^2+3 x+8$
- $y^2=23 x^6+30 x^5+31 x^4+2 x^3+31 x^2+30 x+23$
- $y^2=24 x^6+22 x^5+6 x^4+17 x^3+5 x^2+20 x+19$
- $y^2=23 x^6+2 x^5+30 x^4+26 x^3+12 x^2+24 x+29$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{37}$.
Endomorphism algebra over $\F_{37}$| The isogeny class factors as 1.37.am $\times$ 1.37.ag and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.