Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $3600$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 60.96.1.214 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}11&24\\27&35\end{bmatrix}$, $\begin{bmatrix}17&30\\51&31\end{bmatrix}$, $\begin{bmatrix}41&58\\57&53\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.48.1.u.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: | $2^{4}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3600.2.a.v |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 5 x y - 3 y^{2} + 2 y z - 2 z^{2} $ |
| $=$ | $27 x^{2} - 12 x y - 3 y^{2} + 12 y z - 12 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} - 24 x^{3} z + 3 x^{2} y^{2} - x^{2} z^{2} + 6 x z^{3} - z^{4} $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^6}{3}\cdot\frac{1271133388800xz^{11}-58058726400xz^{9}w^{2}+4481775360xz^{7}w^{4}-75193920xz^{5}w^{6}+628560xz^{3}w^{8}-123309195264y^{2}z^{10}+21892239360y^{2}z^{8}w^{2}-2688650496y^{2}z^{6}w^{4}+185526720y^{2}z^{4}w^{6}-7001712y^{2}z^{2}w^{8}+111972y^{2}w^{10}+172840697856yz^{11}-80396375040yz^{9}w^{2}+6987803904yz^{7}w^{4}-369835200yz^{5}w^{6}+8278128yz^{3}w^{8}-55968yzw^{10}-845358944256z^{12}+26075105280z^{10}w^{2}-3345110784z^{8}w^{4}+38119680z^{6}w^{6}-1083888z^{4}w^{8}-2232z^{2}w^{10}-5w^{12}}{w^{4}(14929920xz^{7}-2941920xz^{5}w^{2}+125280xz^{3}w^{4}+5211648y^{2}z^{6}-1212480y^{2}z^{4}w^{2}+73164y^{2}z^{2}w^{4}-729y^{2}w^{6}-9718272yz^{7}+485280yz^{5}w^{2}+196824yz^{3}w^{4}-11664yzw^{6}-5211648z^{8}+1054080z^{6}w^{2}-47124z^{4}w^{4}+64z^{2}w^{6})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.48.1.u.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{5}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3z$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}+3X^{2}Y^{2}-24X^{3}Z-X^{2}Z^{2}+6XZ^{3}-Z^{4} $ |
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.i.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 30.48.0-30.a.1.1 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-30.a.1.8 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-12.i.1.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.1-60.w.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.48.1-60.w.1.11 | $60$ | $2$ | $2$ | $1$ | $0$ | 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.is.1.3 | $60$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 60.480.17-60.ih.1.1 | $60$ | $5$ | $5$ | $17$ | $8$ | $1^{16}$ |
| 60.576.17-60.dm.1.4 | $60$ | $6$ | $6$ | $17$ | $1$ | $1^{16}$ |
| 60.960.33-60.go.1.9 | $60$ | $10$ | $10$ | $33$ | $13$ | $1^{32}$ |
| 180.288.5-180.u.1.6 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 180.288.9-180.cp.1.7 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 180.288.9-180.ct.1.7 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |