Invariants
| Level: | $60$ | $\SL_2$-level: | $10$ | Newform level: | $900$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot10^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 10D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.24.1.105 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}48&35\\35&51\end{bmatrix}$, $\begin{bmatrix}51&40\\58&23\end{bmatrix}$, $\begin{bmatrix}59&55\\10&7\end{bmatrix}$ |
| 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: | $92160$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 900.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - y w $ |
| $=$ | $11 x^{2} - 25 y^{2} + 11 y w + 15 z^{2} - 5 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 25 x^{4} - 22 x^{2} z^{2} - 15 y^{2} z^{2} + 5 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 between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{3562500yz^{4}w+173964000yz^{2}w^{3}+4376384yw^{5}-78125z^{6}-34365000z^{4}w^{2}-119227920z^{2}w^{4}-1647360w^{6}}{w(140625yz^{4}+450750yz^{2}w^{2}+68381yw^{4}+247500z^{4}w-5280z^{2}w^{3}-25740w^{5})}$ |
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 |
|---|---|---|---|---|---|---|
| 20.12.0.p.2 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 30.12.1.f.1 | $30$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.12.0.bl.2 | $60$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.72.1.cl.1 | $60$ | $3$ | $3$ | $1$ | $1$ | dimension zero |
| 60.72.5.df.1 | $60$ | $3$ | $3$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.96.5.bx.2 | $60$ | $4$ | $4$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.96.5.ck.1 | $60$ | $4$ | $4$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.120.5.ct.1 | $60$ | $5$ | $5$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 300.120.5.p.2 | $300$ | $5$ | $5$ | $5$ | $?$ | not computed |