Invariants
| Base field: | $\F_{5^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 12 x + 72 x^{2} - 300 x^{3} + 625 x^{4}$ |
| Frobenius angles: | $\pm0.0725107809371$, $\pm0.427489219063$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{14})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $14$ |
| Isomorphism classes: | 16 |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $386$ | $389860$ | $243578738$ | $151990819600$ | $95286489448466$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $626$ | $15590$ | $389094$ | $9757334$ | $244140626$ | $6103675550$ | $152588279614$ | $3814695601454$ | $95367431640626$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(4a+1)x^6+(4a+4)x^5+3ax^4+3ax^2+(a+1)x+4a+1$
- $y^2=3ax^6+2x^5+2x^4+(a+2)x^3+(3a+3)x^2+4ax+3a$
- $y^2=(4a+3)x^6+(3a+2)x^5+(2a+2)x^4+x^3+4ax+4a+1$
- $y^2=(3a+4)x^6+(3a+3)x^5+(3a+4)x^4+x^3+2ax^2+(4a+3)x+a+2$
- $y^2=(3a+2)x^6+(4a+1)x^5+(4a+1)x^4+(4a+2)x^3+2ax^2+ax+a+4$
- $y^2=2ax^6+(3a+3)x^5+3x^4+3x^2+(2a+2)x+2a$
- $y^2=(4a+4)x^6+(4a+3)x^5+(4a+2)x^4+(3a+2)x^3+(3a+3)x^2+3x+4a+1$
- $y^2=(2a+2)x^6+(a+3)x^5+2x^4+(2a+1)x^3+(4a+2)x^2+3ax+a$
- $y^2=(4a+3)x^6+ax^5+x^4+3x^3+(a+2)x^2+(4a+1)x+4a+3$
- $y^2=(a+4)x^6+4ax^5+(3a+2)x^4+(4a+3)x^3+(2a+3)x^2+2ax+1$
- $y^2=(4a+1)x^6+(3a+4)x^5+2ax^4+(4a+4)x^3+(3a+1)x^2+2x+2a+4$
- $y^2=4ax^6+2ax^5+(2a+4)x^4+(2a+1)x^3+(3a+1)x^2+x+a+4$
- $y^2=2ax^6+(3a+2)x^5+(3a+1)x^4+(2a+2)x^3+(4a+2)x^2+2a$
- $y^2=(a+4)x^6+(2a+4)x^5+(3a+3)x^4+(4a+2)x^3+(4a+1)x^2+(2a+2)x+a+4$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{8}}$.
Endomorphism algebra over $\F_{5^{2}}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{14})\). |
| The base change of $A$ to $\F_{5^{8}}$ is 1.390625.abdm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-14}) \)$)$ |
- Endomorphism algebra over $\F_{5^{4}}$
The base change of $A$ to $\F_{5^{4}}$ is the simple isogeny class 2.625.a_abdm and its endomorphism algebra is \(\Q(i, \sqrt{14})\).
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.m_cu | $2$ | 2.625.a_abdm |
| 2.25.a_aw | $8$ | (not in LMFDB) |
| 2.25.a_w | $8$ | (not in LMFDB) |