Invariants
| Level: | $60$ | $\SL_2$-level: | $30$ | Newform level: | $3600$ | ||
| Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot30^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30K9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.144.9.150 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}13&20\\37&1\end{bmatrix}$, $\begin{bmatrix}28&5\\19&13\end{bmatrix}$, $\begin{bmatrix}53&50\\29&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $15360$ |
Jacobian
| Conductor: | $2^{34}\cdot3^{14}\cdot5^{13}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{2}$ |
| Newforms: | 240.2.f.b, 400.2.a.c, 720.2.a.c, 720.2.f.d, 900.2.a.b, 3600.2.a.e$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x t + y w $ |
| $=$ | $x v + z u - z v$ | |
| $=$ | $y v + z u - z r$ | |
| $=$ | $w u - w r + t u - t v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 675 x^{4} y^{12} + 900 x^{4} y^{10} z^{2} + 300 x^{4} y^{8} z^{4} - 10935 x^{2} y^{14} + 4050 x^{2} y^{10} z^{4} + \cdots + 1875 z^{16} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle s$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{5}{3}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{5}{3}v$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 60.72.5.ca.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x-y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y-z$ |
| $\displaystyle W$ | $=$ | $\displaystyle w+t$ |
| $\displaystyle T$ | $=$ | $\displaystyle -s$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}-ZW $ |
| $=$ | $ 3X^{2}-5Y^{2}-3XZ-15XW+3ZW $ | |
| $=$ | $ 4X^{2}-10Y^{2}+3Z^{2}+5ZW+75W^{2}-T^{2} $ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 30.72.1.o.1 | $30$ | $2$ | $2$ | $1$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.72.3.yl.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{4}\cdot2$ |
| 60.72.3.bbw.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{4}\cdot2$ |
| 60.72.5.ca.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $2^{2}$ |
| 60.72.5.cp.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.72.5.cs.2 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.72.5.eg.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.432.25.mk.2 | $60$ | $3$ | $3$ | $25$ | $3$ | $1^{8}\cdot2^{4}$ |
| 60.576.41.jw.1 | $60$ | $4$ | $4$ | $41$ | $8$ | $1^{16}\cdot2^{8}$ |
| 60.720.49.cbr.1 | $60$ | $5$ | $5$ | $49$ | $9$ | $1^{18}\cdot2^{11}$ |