Invariants
| Base field: | $\F_{2^{10}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 61 x + 1024 x^{2} )( 1 - 59 x + 1024 x^{2} )$ |
| $1 - 120 x + 5647 x^{2} - 122880 x^{3} + 1048576 x^{4}$ | |
| Frobenius angles: | $\pm0.0978468837242$, $\pm0.126656933887$ |
| Angle rank: | $2$ (numerical) |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $931224$ | $1096259242176$ | $1152853088880563784$ | $1208925096012802153440000$ | $1267650628394869560858818592504$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $905$ | $1045471$ | $1073678105$ | $1099510969663$ | $1125899931859625$ | $1152921506752220191$ | $1180591620819531304505$ | $1208925819618532823801983$ | $1237940039285509432379266505$ | $1267650600228233155406745729631$ |
Jacobians and polarizations
This isogeny class contains a Jacobian, and hence is principally polarizable.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{10}}$.
Endomorphism algebra over $\F_{2^{10}}$| The isogeny class factors as 1.1024.acj $\times$ 1.1024.ach 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.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.1024.ac_achr | $2$ | (not in LMFDB) |
| 2.1024.c_achr | $2$ | (not in LMFDB) |
| 2.1024.eq_ijf | $2$ | (not in LMFDB) |