Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | Newform level: | $576$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16G1 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}31&230\\156&157\end{bmatrix}$, $\begin{bmatrix}97&194\\56&63\end{bmatrix}$, $\begin{bmatrix}99&136\\188&31\end{bmatrix}$, $\begin{bmatrix}124&191\\97&94\end{bmatrix}$, $\begin{bmatrix}207&164\\170&133\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.bo.2 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $1536$ |
| Full 240-torsion field degree: | $5898240$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 396x + 3024 $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot3^2}\cdot\frac{144x^{2}y^{14}+29877984x^{2}y^{12}z^{2}+3421586064384x^{2}y^{10}z^{4}+7019709937815960576x^{2}y^{8}z^{6}-57120375286048493666304x^{2}y^{6}z^{8}-236156864454986300239970304x^{2}y^{4}z^{10}-274890497894311762852615028736x^{2}y^{2}z^{12}-102928938192580688367952267837440x^{2}z^{14}+14544xy^{14}z+17991113472xy^{12}z^{3}+2571483498045696xy^{10}z^{5}-220105323076723642368xy^{8}z^{7}+1101433099070134426927104xy^{6}z^{9}+5188818435801971885029195776xy^{4}z^{11}+6227671554843046892456056455168xy^{2}z^{13}+2364335633386795431420220019834880xz^{15}-y^{16}-3120768y^{14}z^{2}-1123561767168y^{12}z^{4}-179518228013592576y^{10}z^{6}+3440830218852579360768y^{8}z^{8}+1955708659347210781065216y^{6}z^{10}-20327697899449010897212145664y^{4}z^{12}-32204524649248942861882748829696y^{2}z^{14}-13550260500909930790438851991044096z^{16}}{zy^{2}(12060x^{2}y^{10}z-1577968128x^{2}y^{8}z^{3}+56716488707328x^{2}y^{6}z^{5}-827509228120793088x^{2}y^{4}z^{7}+5268744303613522624512x^{2}y^{2}z^{9}-12176669570456349195632640x^{2}z^{11}+xy^{12}-765072xy^{10}z^{2}+64012591872xy^{8}z^{4}-1825817140076544xy^{6}z^{6}+22945477495143960576xy^{4}z^{8}-131284894794440127086592xy^{2}z^{10}+279704952435646619537375232xz^{12}-144y^{12}z+35943264y^{10}z^{3}-1797901277184y^{8}z^{5}+34388360257222656y^{6}z^{7}-298457973959800651776y^{4}z^{9}+1164917259352026520289280y^{2}z^{11}-1603019011082045150277402624z^{13})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 80.48.0-16.e.1.2 | $80$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.48.0-24.by.1.14 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-16.e.1.11 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-24.by.1.5 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.1-48.a.1.20 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.1-48.a.1.27 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 240.192.1-48.m.2.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.u.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bf.2.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.bv.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.dq.2.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.dt.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.ef.2.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-48.ek.2.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.nv.2.13 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.od.2.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.pb.2.13 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.pj.2.12 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.st.2.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.tb.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.tz.2.13 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.uh.2.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.288.9-48.iq.2.14 | $240$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 240.384.9-48.bfh.1.27 | $240$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.480.17-240.ek.1.3 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |