Invariants
| Level: | $40$ | $\SL_2$-level: | $10$ | Newform level: | $1600$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 40.12.1.7 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&15\\30&13\end{bmatrix}$, $\begin{bmatrix}23&28\\30&1\end{bmatrix}$, $\begin{bmatrix}27&32\\39&25\end{bmatrix}$, $\begin{bmatrix}32&9\\15&36\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $61440$ |
Jacobian
| Conductor: | $2^{6}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1600.2.a.w |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} - 3633x + 129137 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(-73:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 12 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^5\cdot5^5}\cdot\frac{140x^{2}y^{2}+10000x^{2}z^{2}+9340xy^{2}z-173540000xz^{3}+y^{4}-56240y^{2}z^{2}-12596710000z^{4}}{z^{3}(x+73z)}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
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 |
| 40.2.0.a.1 | $40$ | $6$ | $6$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.24.1.cj.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.cj.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.ck.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.ck.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.cm.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.cm.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.cn.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.cn.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.36.1.e.1 | $40$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 40.48.3.g.1 | $40$ | $4$ | $4$ | $3$ | $1$ | $1^{2}$ |
| 40.60.3.bd.1 | $40$ | $5$ | $5$ | $3$ | $1$ | $1^{2}$ |
| 120.24.1.iz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.iz.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.ja.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.ja.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.jc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.jc.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.jd.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.jd.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.3.e.1 | $120$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 120.48.3.ci.1 | $120$ | $4$ | $4$ | $3$ | $?$ | not computed |
| 200.60.3.c.1 | $200$ | $5$ | $5$ | $3$ | $?$ | not computed |
| 280.24.1.cj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.cj.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.ck.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.ck.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.cm.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.cm.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.cn.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.cn.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.7.c.1 | $280$ | $8$ | $8$ | $7$ | $?$ | not computed |
| 280.252.19.c.1 | $280$ | $21$ | $21$ | $19$ | $?$ | not computed |