Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $1$ |
L-polynomial: | $1 + 10 x + 169 x^{2}$ |
Frobenius angles: | $\pm0.625665916378$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-1}) \) |
Galois group: | $C_2$ |
Jacobians: | $18$ |
This isogeny class is simple and geometrically simple, not primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $1$ |
Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $180$ | $28800$ | $4822740$ | $815731200$ | $137859174900$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $180$ | $28800$ | $4822740$ | $815731200$ | $137859174900$ | $23298078211200$ | $3937376339376660$ | $665416610814412800$ | $112455406943473588020$ | $19004963774689959120000$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:
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 \(\Q(\sqrt{-1}) \). |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{13^{2}}$.
Subfield | Primitive Model |
$\F_{13}$ | 1.13.ae |
$\F_{13}$ | 1.13.e |
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
1.169.ak | $2$ | (not in LMFDB) |
1.169.ay | $4$ | (not in LMFDB) |
1.169.y | $4$ | (not in LMFDB) |