Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $1800$ | ||
| 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: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.96.1.108 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}9&56\\38&9\end{bmatrix}$, $\begin{bmatrix}27&56\\28&1\end{bmatrix}$, $\begin{bmatrix}43&42\\18&1\end{bmatrix}$, $\begin{bmatrix}55&52\\36&47\end{bmatrix}$, $\begin{bmatrix}59&56\\4&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.48.1.d.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $192$ |
| Full 60-torsion field degree: | $23040$ |
Jacobian
| Conductor: | $2^{3}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1800.2.a.m |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 975x - 8750 $ |
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^6\cdot5^6}\cdot\frac{240x^{2}y^{14}+119981250x^{2}y^{12}z^{2}+11699083125000x^{2}y^{10}z^{4}+698141978173828125x^{2}y^{8}z^{6}+25165314255322265625000x^{2}y^{6}z^{8}+619858653753626861572265625x^{2}y^{4}z^{10}+9019146589735154342651367187500x^{2}y^{2}z^{12}+76569767265118828289508819580078125x^{2}z^{14}+27300xy^{14}z+6773625000xy^{12}z^{3}+585484674609375xy^{10}z^{5}+31320840931347656250xy^{8}z^{7}+1064641412240332031250000xy^{6}z^{9}+24485121511674499511718750000xy^{4}z^{11}+349143395563227970218658447265625xy^{2}z^{13}+2661114036651611717104911804199218750xz^{15}+y^{16}+1950000y^{14}z^{2}+247784062500y^{12}z^{4}+17154964687500000y^{10}z^{6}+714799505012695312500y^{8}z^{8}+20529202021268554687500000y^{6}z^{10}+375035249606280372619628906250y^{4}z^{12}+4263176617176149677276611328125000y^{2}z^{14}+18954175912072025246679782867431640625z^{16}}{z^{4}y^{4}(180x^{2}y^{6}+46524375x^{2}y^{4}z^{2}+2215704375000x^{2}y^{2}z^{4}+28341790048828125x^{2}z^{6}+15750xy^{6}z+2300400000xy^{4}z^{3}+88070319140625xy^{2}z^{5}+992040499511718750xz^{7}+y^{8}+882000y^{6}z^{2}+62410500000y^{4}z^{4}+1289219414062500y^{2}z^{6}+7087393707275390625z^{8})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-6.a.1.9 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-6.a.1.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-60.t.1.8 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-60.t.1.9 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.1-60.v.1.8 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.48.1-60.v.1.9 | $60$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 60.192.1-60.h.1.8 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.192.1-60.h.2.7 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.192.1-60.h.3.7 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.192.1-60.h.4.5 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.192.3-60.e.1.7 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.192.3-60.f.1.12 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.192.3-60.h.1.11 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.192.3-60.i.1.12 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.192.3-60.r.1.12 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.192.3-60.r.2.6 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.192.3-60.v.1.12 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.192.3-60.v.2.6 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.288.5-60.e.1.6 | $60$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 60.480.17-60.h.1.14 | $60$ | $5$ | $5$ | $17$ | $3$ | $1^{16}$ |
| 60.576.17-60.h.1.36 | $60$ | $6$ | $6$ | $17$ | $5$ | $1^{16}$ |
| 60.960.33-60.t.1.34 | $60$ | $10$ | $10$ | $33$ | $5$ | $1^{32}$ |
| 120.192.1-120.lw.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lw.2.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lw.3.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lw.4.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-120.dy.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eb.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eh.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ek.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fu.1.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fu.2.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.gn.1.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.gn.2.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.288.5-180.d.1.3 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 180.288.9-180.d.1.11 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 180.288.9-180.h.1.4 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |