Invariants
Base field: | $\F_{163}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 48 x + 900 x^{2} - 7824 x^{3} + 26569 x^{4}$ |
Frobenius angles: | $\pm0.0308818759184$, $\pm0.154480826596$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.55552.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $19598$ | $692632516$ | $18736047682094$ | $498290274138211216$ | $13239625854229675403198$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $116$ | $26066$ | $4326284$ | $705881814$ | $115063529156$ | $18755370777026$ | $3057125262158300$ | $498311414352315358$ | $81224760528362714708$ | $13239635966855815968146$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=41x^6+85x^5+39x^4+160x^3+30x^2+101x+4$
- $y^2=123x^6+140x^5+63x^4+143x^3+59x^2+48x+19$
- $y^2=138x^6+140x^5+121x^4+52x^3+39x^2+159x+75$
- $y^2=92x^6+28x^5+122x^4+156x^3+7x^2+85x+111$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{163}$.
Endomorphism algebra over $\F_{163}$The endomorphism algebra of this simple isogeny class is 4.0.55552.1. |
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 |
---|---|---|
2.163.bw_biq | $2$ | (not in LMFDB) |