Invariants
| Level: | $60$ | $\SL_2$-level: | $10$ | Newform level: | $900$ | ||
| Index: | $40$ | $\PSL_2$-index: | $40$ | ||||
| Genus: | $1 = 1 + \frac{ 40 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $10^{4}$ | Cusp orbits | $4$ | ||
| Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.40.1.40 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}38&21\\53&17\end{bmatrix}$, $\begin{bmatrix}47&3\\8&43\end{bmatrix}$, $\begin{bmatrix}47&47\\37&18\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $144$ |
| Cyclic 60-torsion field degree: | $2304$ |
| Full 60-torsion field degree: | $55296$ |
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 z - 2 z^{2} + w^{2} $ |
| $=$ | $3 x^{2} + 3 y^{2} + y z + z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} + 90 x^{2} y^{2} - 5 x^{2} z^{2} + 441 y^{4} - 42 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 between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 40 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\cdot5^2\,\frac{(4z^{2}-3w^{2})(944yz^{7}-5748yz^{5}w^{2}+8352yz^{3}w^{4}-3456yzw^{6}-2848z^{8}+8036z^{6}w^{2}-4509z^{4}w^{4}-2808z^{2}w^{6}+2160w^{8})}{1475yz^{9}-1650yz^{7}w^{2}-675yz^{5}w^{4}+1350yz^{3}w^{6}-405yzw^{8}-4450z^{10}+11675z^{8}w^{2}-11400z^{6}w^{4}+4725z^{4}w^{6}-540z^{2}w^{8}-81w^{10}}$ |
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 |
|---|---|---|---|---|---|---|
| 20.20.0.b.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 30.20.1.a.1 | $30$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.20.0.c.1 | $60$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.120.5.x.1 | $60$ | $3$ | $3$ | $5$ | $3$ | $1^{4}$ |
| 60.120.5.bh.1 | $60$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 60.120.7.cg.1 | $60$ | $3$ | $3$ | $7$ | $5$ | $1^{4}\cdot2$ |
| 60.120.9.cf.1 | $60$ | $3$ | $3$ | $9$ | $4$ | $1^{6}\cdot2$ |
| 60.160.9.bb.1 | $60$ | $4$ | $4$ | $9$ | $4$ | $1^{8}$ |
| 60.160.9.bf.1 | $60$ | $4$ | $4$ | $9$ | $6$ | $1^{8}$ |
| 300.200.9.i.1 | $300$ | $5$ | $5$ | $9$ | $?$ | not computed |