Invariants
| Level: | $60$ | $\SL_2$-level: | $30$ | Newform level: | $900$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot6^{2}\cdot10^{2}\cdot30^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30N5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.192.5.1005 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}11&5\\33&52\end{bmatrix}$, $\begin{bmatrix}11&55\\15&26\end{bmatrix}$, $\begin{bmatrix}41&30\\39&53\end{bmatrix}$, $\begin{bmatrix}59&15\\0&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.96.5.g.2 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $6$ |
| Cyclic 60-torsion field degree: | $96$ |
| Full 60-torsion field degree: | $11520$ |
Jacobian
| Conductor: | $2^{8}\cdot3^{9}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 15.2.a.a, 180.2.d.a, 900.2.a.b$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 2 y z - y w - y t + 6 z^{2} + 2 z w - z t - w t - t^{2} $ |
| $=$ | $y^{2} - 2 y z - 6 y w + y t - 3 z w - w t$ | |
| $=$ | $15 x^{2} + y z + 2 y w + 2 y t - z w - 2 w^{2} - 2 w t$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{8} - 300 x^{6} y^{2} + 4500 x^{4} y^{4} - 6 x^{4} y^{2} z^{2} + 300 x^{2} y^{4} z^{2} + \cdots + y^{4} z^{4} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:1/2:0:1)$, $(0:0:-1/3:0:1)$, $(0:-5:3:-5/2:1)$, $(0:-5/7:6/7:-10/7:1)$ |
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 30.48.3.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -x+z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x-y-2z-w$ |
Equation of the image curve:
| $0$ | $=$ | $ 36X^{4}-10X^{3}Y-9X^{2}Y^{2}-2XY^{3}-2X^{3}Z+12X^{2}YZ-2Y^{3}Z+3X^{2}Z^{2}+12XYZ^{2}+3Y^{2}Z^{2}+2XZ^{3}+2YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 30.96.5.g.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{15}y-\frac{1}{15}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 5z$ |
Equation of the image curve:
| $0$ | $=$ | $ 5X^{8}-300X^{6}Y^{2}+4500X^{4}Y^{4}-6X^{4}Y^{2}Z^{2}+300X^{2}Y^{4}Z^{2}-8100Y^{6}Z^{2}+Y^{4}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.48.1-30.i.2.4 | $60$ | $4$ | $4$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 60.96.1-15.a.2.4 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 60.96.1-15.a.2.13 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.384.9-30.f.2.4 | $60$ | $2$ | $2$ | $9$ | $2$ | $4$ |
| 60.384.9-30.f.4.4 | $60$ | $2$ | $2$ | $9$ | $2$ | $4$ |
| 60.384.9-30.h.3.3 | $60$ | $2$ | $2$ | $9$ | $2$ | $4$ |
| 60.384.9-30.h.4.2 | $60$ | $2$ | $2$ | $9$ | $2$ | $4$ |
| 60.384.9-60.v.1.3 | $60$ | $2$ | $2$ | $9$ | $2$ | $4$ |
| 60.384.9-60.v.2.3 | $60$ | $2$ | $2$ | $9$ | $2$ | $4$ |
| 60.384.9-60.bb.3.1 | $60$ | $2$ | $2$ | $9$ | $2$ | $4$ |
| 60.384.9-60.bb.4.1 | $60$ | $2$ | $2$ | $9$ | $2$ | $4$ |
| 60.576.13-30.g.1.1 | $60$ | $3$ | $3$ | $13$ | $2$ | $1^{4}\cdot2^{2}$ |
| 60.576.17-30.i.2.2 | $60$ | $3$ | $3$ | $17$ | $3$ | $1^{6}\cdot2^{3}$ |
| 60.768.25-60.ck.1.1 | $60$ | $4$ | $4$ | $25$ | $7$ | $1^{10}\cdot2^{5}$ |
| 60.960.29-30.s.1.6 | $60$ | $5$ | $5$ | $29$ | $5$ | $1^{12}\cdot2^{6}$ |
| 120.384.9-120.dbn.1.7 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.dbn.2.7 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.dbq.1.3 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.dbq.2.3 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.dcl.3.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.dcl.4.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.dco.3.3 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.dco.4.3 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |