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: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.96.1.323 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}19&42\\36&29\end{bmatrix}$, $\begin{bmatrix}35&18\\13&35\end{bmatrix}$, $\begin{bmatrix}35&48\\44&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.48.1.s.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $96$ |
| 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
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - y^{2} + z^{2} $ |
| $=$ | $12 x^{2} + 3 y^{2} - 8 z^{2} + 3 w^{2}$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^4}{3^2}\cdot\frac{(20z^{2}-3w^{2})(349440000xyz^{8}-131040000xyz^{6}w^{2}+2462400xyz^{4}w^{4}-110160xyz^{2}w^{6}-88452xyw^{8}-233600000z^{10}+175200000z^{8}w^{2}-29232000z^{6}w^{4}+885600z^{4}w^{6}-66420z^{2}w^{8}+17739w^{10})}{w^{4}z^{2}(24000xyz^{4}-1800xyz^{2}w^{2}-270xyw^{4}+16000z^{6}-7200z^{4}w^{2}-135z^{2}w^{4}-54w^{6})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-12.h.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-12.h.1.2 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-60.p.1.5 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-60.p.1.9 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.1-60.v.1.5 | $60$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 60.288.5-60.iy.1.4 | $60$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 60.480.17-60.if.1.2 | $60$ | $5$ | $5$ | $17$ | $9$ | $1^{16}$ |
| 60.576.17-60.dk.1.3 | $60$ | $6$ | $6$ | $17$ | $5$ | $1^{16}$ |
| 60.960.33-60.gm.1.13 | $60$ | $10$ | $10$ | $33$ | $15$ | $1^{32}$ |
| 180.288.5-180.s.1.7 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 180.288.9-180.cl.1.6 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 180.288.9-180.cn.1.5 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |