Invariants
| Base field: | $\F_{5^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 14 x + 97 x^{2} - 350 x^{3} + 625 x^{4}$ |
| Frobenius angles: | $\pm0.181719349551$, $\pm0.311346897918$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.128576.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $359$ | $390233$ | $248555804$ | $153258937721$ | $95415159378519$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $624$ | $15906$ | $392340$ | $9770512$ | $244142022$ | $6103486128$ | $152587981092$ | $3814698828306$ | $95367433546144$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(4a+1)x^6+(2a+1)x^5+(2a+1)x^4+(3a+1)x^3+(3a+3)x^2+(a+3)x+a+4$
- $y^2=3ax^6+(3a+2)x^5+(4a+2)x^4+(4a+2)x^3+(4a+4)x^2+2x+4a+3$
- $y^2=(a+4)x^6+(3a+4)x^5+(2a+2)x^4+(4a+2)x^2+(4a+3)x+3a$
- $y^2=3x^6+3x^5+(a+1)x^4+ax^3+(3a+3)x^2+4ax+4a+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5^{2}}$| The endomorphism algebra of this simple isogeny class is 4.0.128576.2. |
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 |
|---|---|---|
| 2.25.o_dt | $2$ | 2.625.ac_bhb |