Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.1.306 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&28\\6&25\end{bmatrix}$, $\begin{bmatrix}17&38\\24&23\end{bmatrix}$, $\begin{bmatrix}25&6\\24&19\end{bmatrix}$, $\begin{bmatrix}35&18\\24&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.1.a.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $15360$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 4x $ |
Rational points
This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | $(0:1:0)$, $(0:0:1)$ | ||
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(2:-4:1)$, $(2:4:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{48x^{2}y^{4}z^{2}-4xy^{6}z+768xy^{2}z^{5}+y^{8}+4096z^{8}}{z^{2}y^{4}x^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.24.0-4.a.1.2 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0-4.a.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.96.1-8.b.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-8.c.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.e.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-8.f.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.f.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-8.h.2.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.i.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.j.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.240.9-40.a.1.7 | $40$ | $5$ | $5$ | $9$ | $2$ | $1^{6}\cdot2$ |
| 40.288.9-40.a.1.12 | $40$ | $6$ | $6$ | $9$ | $1$ | $1^{6}\cdot2$ |
| 40.480.17-40.m.1.15 | $40$ | $10$ | $10$ | $17$ | $5$ | $1^{12}\cdot2^{2}$ |
| 120.96.1-24.e.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.e.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-24.f.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.f.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-24.i.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.i.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-24.j.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.j.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.5-24.a.1.13 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.192.5-24.a.1.9 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.96.1-56.e.1.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.e.1.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-56.f.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.f.1.5 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-56.i.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.i.1.5 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-56.j.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1-280.j.1.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.384.13-56.a.1.15 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |