Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | Newform level: | $48$ | ||
| 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: | $\begin{bmatrix}9&4\\16&57\end{bmatrix}$, $\begin{bmatrix}15&74\\56&75\end{bmatrix}$, $\begin{bmatrix}25&62\\34&51\end{bmatrix}$, $\begin{bmatrix}33&62\\68&93\end{bmatrix}$, $\begin{bmatrix}43&50\\40&51\end{bmatrix}$, $\begin{bmatrix}93&94\\34&63\end{bmatrix}$, $\begin{bmatrix}117&104\\2&75\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.48.1.a.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: | 48.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 4x - 4 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{16x^{2}y^{14}+254370x^{2}y^{12}z^{2}+272792952x^{2}y^{10}z^{4}+43885840685x^{2}y^{8}z^{6}+1881465528280x^{2}y^{6}z^{8}+31188954184569x^{2}y^{4}z^{10}+218165771777620x^{2}y^{2}z^{12}+541652977285693x^{2}z^{14}+852xy^{14}z+2923620xy^{12}z^{3}+2032924815xy^{10}z^{5}+217907337732xy^{8}z^{7}+7546917207824xy^{6}z^{9}+109553117422806xy^{4}z^{11}+699635304814401xy^{2}z^{13}+1624959302500352xz^{15}+y^{16}+11420y^{14}z^{2}+33883710y^{12}z^{4}+9961460717y^{10}z^{6}+628484953611y^{8}z^{8}+14424778023672y^{6}z^{10}+146070717545179y^{4}z^{12}+662020649013103y^{2}z^{14}+1083306712635148z^{16}}{zy^{4}(77x^{2}y^{8}z-128x^{2}y^{6}z^{3}-1280x^{2}y^{4}z^{5}-2048x^{2}y^{2}z^{7}+16384x^{2}z^{9}+xy^{10}+224xy^{8}z^{2}+384xy^{6}z^{4}+2048xy^{4}z^{6}+6144xy^{2}z^{8}-16384xz^{10}+15y^{10}z-20y^{8}z^{3}+1280y^{4}z^{7}-8192y^{2}z^{9}-32768z^{11})}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 40.24.0-4.a.1.1 | $40$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.24.0-4.a.1.1 | $40$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 120.48.0-6.a.1.7 | $120$ | $2$ | $2$ | $0$ | $?$ | 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.a.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-12.a.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-12.a.2.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-12.a.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.a.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.a.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.a.2.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.a.2.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.ci.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.ci.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.ci.2.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.ci.2.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.le.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.le.1.17 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.le.2.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.le.2.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-12.a.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.a.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.b.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.b.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.c.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.d.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.d.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.d.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.e.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.e.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.f.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.f.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.f.2.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.f.2.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.k.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.k.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.k.2.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.k.2.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bb.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bd.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bi.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bi.2.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bj.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bj.2.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bs.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bs.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bs.2.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bs.2.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.ce.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.cf.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.ci.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.cj.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.do.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.dq.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.dv.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.dx.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eo.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eo.2.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ep.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ep.2.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ey.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ey.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ey.2.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ey.2.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fw.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fx.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ga.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.gb.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.5-24.a.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.a.1.17 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-24.b.1.19 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.b.1.40 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-24.e.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.e.1.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-24.f.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.f.1.34 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-24.i.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-24.i.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.i.1.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.i.2.16 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-24.j.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-24.j.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.j.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.j.2.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.a.1.4 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.480.17-60.a.1.7 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |