Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 49 x + 937 x^{2} - 8281 x^{3} + 28561 x^{4}$ |
Frobenius angles: | $\pm0.0546290859232$, $\pm0.144071978349$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.63725.1 |
Galois group: | $D_{4}$ |
Jacobians: | $3$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21169$ | $800802101$ | $23275156499641$ | $665391630093649349$ | $19004962180730770132624$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $121$ | $28035$ | $4822057$ | $815700099$ | $137858480286$ | $23298088941651$ | $3937376465835409$ | $665416610310150339$ | $112455406964048221873$ | $19004963774971806607150$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(4a+3)x^6+(5a+7)x^5+(10a+12)x^4+(10a+9)x^3+(7a+10)x^2+(6a+5)x+9a$
- $y^2=(10a+10)x^6+(a+1)x^5+(7a+11)x^4+(11a+2)x^3+(3a+5)x^2+(9a+1)x+9a$
- $y^2=(3a+9)x^6+(2a+5)x^5+(3a+11)x^4+(12a+10)x^3+(10a+1)x^2+(11a+10)x+4a+11$
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 4.0.63725.1. |
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.bx_bkb | $2$ | (not in LMFDB) |