Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 47 x + 886 x^{2} - 7943 x^{3} + 28561 x^{4}$ |
Frobenius angles: | $\pm0.0585395387306$, $\pm0.191425177757$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1413788.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21458$ | $803344604$ | $23284933090016$ | $665416595154354368$ | $19005003162788553065538$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $123$ | $28125$ | $4824084$ | $815730705$ | $137858777563$ | $23298089394594$ | $3937376411069091$ | $665416608860373153$ | $112455406940080971684$ | $19004963774688110895805$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(6a+9)x^6+(6a+11)x^5+(6a+1)x^4+11x^3+(2a+5)x^2+(a+4)x+12a+3$
- $y^2=(11a+1)x^6+(2a+11)x^5+12x^4+8x^3+(3a+8)x^2+(12a+3)x+7a$
- $y^2=12ax^6+(7a+3)x^5+(6a+1)x^4+(5a+5)x^3+(2a+12)x^2+(7a+4)x+a+9$
- $y^2=(11a+12)x^6+(2a+3)x^5+(3a+12)x^4+(12a+1)x^3+(3a+1)x^2+9ax+2a+11$
- $y^2=(11a+5)x^6+11x^5+(10a+5)x^4+(8a+11)x^3+(8a+10)x^2+(12a+7)x+2$
- $y^2=(12a+2)x^6+(6a+8)x^5+(12a+6)x^4+(11a+11)x^3+(10a+8)x^2+5ax+4a$
- $y^2=(4a+6)x^6+(7a+6)x^5+(3a+4)x^4+12ax^3+11ax^2+12x+4a+10$
- $y^2=(5a+8)x^6+(11a+8)x^5+(11a+6)x^4+(3a+6)x^3+(11a+5)x^2+(a+1)x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{2}}$.
Endomorphism algebra over $\F_{13^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.1413788.1. |
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 |
---|---|---|
2.169.bv_bic | $2$ | (not in LMFDB) |