Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| 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 $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}13&54\\58&101\end{bmatrix}$, $\begin{bmatrix}19&62\\72&17\end{bmatrix}$, $\begin{bmatrix}35&52\\38&45\end{bmatrix}$, $\begin{bmatrix}37&20\\76&9\end{bmatrix}$, $\begin{bmatrix}49&116\\24&23\end{bmatrix}$, $\begin{bmatrix}73&116\\46&99\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.48.1.c.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $24$ |
| Cyclic 120-torsion field degree: | $768$ |
| Full 120-torsion field degree: | $368640$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 39x + 70 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^6}\cdot\frac{48x^{2}y^{14}-191970x^{2}y^{12}z^{2}+149748264x^{2}y^{10}z^{4}-71489738565x^{2}y^{8}z^{6}+20615425437960x^{2}y^{6}z^{8}-4062305673239769x^{2}y^{4}z^{10}+472863032723906460x^{2}y^{2}z^{12}-32115688111915696197x^{2}z^{14}-1092xy^{14}z+2167560xy^{12}z^{3}-1498840767xy^{10}z^{5}+641450822274xy^{8}z^{7}-174430848981456xy^{6}z^{9}+32093138467782000xy^{4}z^{11}-3661033851461073321xy^{2}z^{13}+223230424967680032630xz^{15}-y^{16}+15600y^{14}z^{2}-15858180y^{12}z^{4}+8783341920y^{10}z^{6}-2927818772532y^{8}z^{8}+672700891832928y^{6}z^{10}-98313240472788762y^{4}z^{12}+8940529369064196648y^{2}z^{14}-317998303378829375121z^{16}}{z^{4}y^{4}(36x^{2}y^{6}-74439x^{2}y^{4}z^{2}+28361016x^{2}y^{2}z^{4}-2902199301x^{2}z^{6}-630xy^{6}z+736128xy^{4}z^{3}-225460017xy^{2}z^{5}+20316989430xz^{7}-y^{8}+7056y^{6}z^{2}-3994272y^{4}z^{4}+660080340y^{2}z^{6}-29029964625z^{8})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 120.24.0-12.a.1.3 | $120$ | $4$ | $4$ | $0$ | $?$ | full Jacobian |
| 120.48.0-6.a.1.7 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.48.0-6.a.1.11 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 120.192.1-12.c.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-12.c.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-12.c.2.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-12.c.2.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.c.1.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.c.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.c.2.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.c.2.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.cm.1.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.cm.1.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.cm.2.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.cm.2.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.li.1.24 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.li.1.25 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.li.2.17 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.li.2.32 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-12.c.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.e.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.e.1.31 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.g.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.g.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.h.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.h.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.h.2.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.h.2.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.h.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.h.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.m.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.m.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.m.2.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.m.2.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bc.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bc.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bg.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bg.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.by.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.by.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.by.2.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.by.2.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ec.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ec.1.29 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ef.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ef.1.25 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fe.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fe.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fe.2.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fe.2.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.5-12.f.1.6 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.480.17-60.c.1.12 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |