Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $1200$ | ||
| 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: | 60.96.1.358 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}3&14\\17&33\end{bmatrix}$, $\begin{bmatrix}29&48\\21&5\end{bmatrix}$, $\begin{bmatrix}41&36\\5&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.48.1.bd.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: | $2^{4}\cdot3\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1200.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 4 x^{2} - y^{2} - z^{2} $ |
| $=$ | $2 x^{2} - x z + 2 x w - y^{2} + 2 z^{2} - 2 z w + 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 24 x^{4} - 18 x^{3} z - 11 x^{2} y^{2} + 13 x^{2} z^{2} + 6 x y^{2} z - 4 x z^{3} + y^{4} - 3 y^{2} z^{2} + 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{z^{3}(12208xz^{8}-105824xz^{7}w+425312xz^{6}w^{2}-1006592xz^{5}w^{3}+1576960xz^{4}w^{4}-1675264xz^{3}w^{5}+1218560xz^{2}w^{6}-557056xzw^{7}+139264xw^{8}-11479z^{9}+105824z^{8}w-426696z^{7}w^{2}+1048368z^{6}w^{3}-1722688z^{5}w^{4}+2016000z^{4}w^{5}-1683200z^{3}w^{6}+1002496z^{2}w^{7}-387072zw^{8}+86016w^{9})}{200xz^{11}-1960xz^{10}w+9274xz^{9}w^{2}-27944xz^{8}w^{3}+59590xz^{7}w^{4}-94564xz^{6}w^{5}+114268xz^{5}w^{6}-105632xz^{4}w^{7}+73712xz^{3}w^{8}-37440xz^{2}w^{9}+12672xzw^{10}-2304xw^{11}-200z^{12}+1960z^{11}w-9354z^{10}w^{2}+28696z^{9}w^{3}-62986z^{8}w^{4}+104268z^{7}w^{5}-133733z^{6}w^{6}+134330z^{5}w^{7}-105453z^{4}w^{8}+63568z^{3}w^{9}-28344z^{2}w^{10}+8544zw^{11}-1424w^{12}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.48.1.bd.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Equation of the image curve:
| $0$ | $=$ | $ 24X^{4}-11X^{2}Y^{2}+Y^{4}-18X^{3}Z+6XY^{2}Z+13X^{2}Z^{2}-3Y^{2}Z^{2}-4XZ^{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.1.7 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-12.i.1.2 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-30.b.1.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-30.b.1.4 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.1-60.y.1.4 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.48.1-60.y.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.lm.1.2 | $60$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 60.480.17-60.kt.1.3 | $60$ | $5$ | $5$ | $17$ | $6$ | $1^{16}$ |
| 60.576.17-60.gf.1.2 | $60$ | $6$ | $6$ | $17$ | $7$ | $1^{16}$ |
| 60.960.33-60.lj.1.13 | $60$ | $10$ | $10$ | $33$ | $11$ | $1^{32}$ |
| 180.288.5-180.bd.1.8 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 180.288.9-180.dh.1.5 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 180.288.9-180.dl.1.5 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |