Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | Newform level: | $192$ | ||
| 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: | $\begin{bmatrix}1&20\\94&69\end{bmatrix}$, $\begin{bmatrix}15&32\\116&15\end{bmatrix}$, $\begin{bmatrix}41&64\\92&27\end{bmatrix}$, $\begin{bmatrix}89&112\\14&93\end{bmatrix}$, $\begin{bmatrix}115&68\\46&87\end{bmatrix}$, $\begin{bmatrix}119&78\\68&103\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.1.bx.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: | 192.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} - 17x - 15 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{32x^{2}y^{14}+37920x^{2}y^{12}z^{2}+8764416x^{2}y^{10}z^{4}+1239741440x^{2}y^{8}z^{6}+105926819840x^{2}y^{6}z^{8}+6184611938304x^{2}y^{4}z^{10}+213305199165440x^{2}y^{2}z^{12}+4292492375621632x^{2}z^{14}+464xy^{14}z+260160xy^{12}z^{3}+52639488xy^{10}z^{5}+6589331456xy^{8}z^{7}+526892793856xy^{6}z^{9}+28450152382464xy^{4}z^{11}+958774337601536xy^{2}z^{13}+17029238129426432xz^{15}+y^{16}+4464y^{14}z^{2}+1301280y^{12}z^{4}+209954560y^{10}z^{6}+20231521280y^{8}z^{8}+1348818370560y^{6}z^{10}+56352057131008y^{4}z^{12}+1449156919951360y^{2}z^{14}+12736757984395264z^{16}}{z^{4}y^{4}(24x^{2}y^{6}+14704x^{2}y^{4}z^{2}+1659904x^{2}y^{2}z^{4}+50328576x^{2}z^{6}+264xy^{6}z+87136xy^{4}z^{3}+7690496xy^{2}z^{5}+201332736xz^{7}+y^{8}+2000y^{6}z^{2}+319984y^{4}z^{4}+14422272y^{2}z^{6}+151041024z^{8})}$ |
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 |
| 40.24.0-8.b.1.2 | $40$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.24.0-8.b.1.2 | $40$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 60.48.0-6.a.1.2 | $60$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 120.192.1-24.cl.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.cl.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.cl.2.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.cl.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.cl.3.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.cl.3.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.cl.4.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.cl.4.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lh.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lh.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lh.2.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lh.2.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lh.3.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lh.3.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lh.4.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lh.4.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-24.e.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.e.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.f.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.f.1.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bg.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bg.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bh.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bh.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bw.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bw.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bw.2.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bw.2.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bx.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bx.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bx.2.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bx.2.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.dt.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.dt.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.du.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.du.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ea.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ea.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eb.1.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eb.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fc.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fc.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fc.2.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fc.2.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fd.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fd.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fd.2.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fd.2.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.5-24.d.1.4 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.480.17-120.fh.1.15 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |