Invariants
| Level: | $60$ | $\SL_2$-level: | $10$ | Newform level: | $3600$ | ||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $1 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot10$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.12.1.6 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}13&15\\44&1\end{bmatrix}$, $\begin{bmatrix}28&15\\29&13\end{bmatrix}$, $\begin{bmatrix}41&55\\15&22\end{bmatrix}$, $\begin{bmatrix}57&5\\59&22\end{bmatrix}$ |
| 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: | $184320$ |
Jacobian
| Conductor: | $2^{4}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3600.2.a.be |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 8175x - 431750 $ |
Rational points
This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3\cdot5}\cdot\frac{10590x^{2}y^{2}+11909272100625x^{2}z^{2}+38307225xy^{2}z+1399749479268750xz^{3}+y^{4}+50486129250y^{2}z^{2}+49539653192953125z^{4}}{x^{2}y^{2}-17374500x^{2}z^{2}-550xy^{2}z+1550593125xz^{3}+129175y^{2}z^{2}+39666206250z^{4}}$ |
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 |
|---|---|---|---|---|---|---|
| $X_0(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.2.0.a.1 | $60$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.24.1.bd.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.24.1.bd.2 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.24.1.be.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.24.1.be.2 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.24.1.bg.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.24.1.bg.2 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.24.1.bh.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.24.1.bh.2 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.36.1.bf.1 | $60$ | $3$ | $3$ | $1$ | $1$ | dimension zero |
| 60.36.3.d.1 | $60$ | $3$ | $3$ | $3$ | $1$ | $1^{2}$ |
| 60.48.3.bb.1 | $60$ | $4$ | $4$ | $3$ | $2$ | $1^{2}$ |
| 60.48.3.bf.1 | $60$ | $4$ | $4$ | $3$ | $2$ | $1^{2}$ |
| 60.60.3.z.1 | $60$ | $5$ | $5$ | $3$ | $2$ | $1^{2}$ |
| 120.24.1.jq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.jq.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.jt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.jt.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.kc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.kc.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.kf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.kf.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 300.60.3.b.1 | $300$ | $5$ | $5$ | $3$ | $?$ | not computed |