Invariants
| Level: | $40$ | $\SL_2$-level: | $10$ | Newform level: | $1600$ | ||
| 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 | $2^{2}\cdot10^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 10D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.1.507 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}0&29\\39&20\end{bmatrix}$, $\begin{bmatrix}6&17\\27&6\end{bmatrix}$, $\begin{bmatrix}25&22\\39&33\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.24.1.cp.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $15360$ |
Jacobian
| Conductor: | $2^{6}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1600.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 4133x - 103637 $ |
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 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2\cdot5}\cdot\frac{7280x^{2}y^{6}+6150536400000x^{2}y^{4}z^{2}+15649414055440000000x^{2}y^{2}z^{4}+4579162597656240000000000x^{2}z^{6}+18612320xy^{6}z+1187760448800000xy^{4}z^{3}+1541479492441760000000xy^{2}z^{5}+342987060546875360000000000xz^{7}+y^{8}+18913549520y^{6}z^{2}+146611209496000000y^{4}z^{4}+76900976582267760000000y^{2}z^{6}+6421647644042996760000000000z^{8}}{y^{2}(x^{2}y^{4}+22000x^{2}y^{2}z^{2}-1000000x^{2}z^{4}-146xy^{4}z-782000xy^{2}z^{3}+36000000xz^{5}+5129y^{4}z^{2}-62352000y^{2}z^{4}+2801000000z^{6})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.24.0-5.a.2.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0-5.a.2.5 | $40$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.144.1-40.bf.2.1 | $40$ | $3$ | $3$ | $1$ | $1$ | dimension zero |
| 40.192.5-40.t.1.8 | $40$ | $4$ | $4$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.240.5-40.cu.1.3 | $40$ | $5$ | $5$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 120.144.5-120.bbf.1.28 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.192.5-120.nj.1.25 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 200.240.5-200.t.2.4 | $200$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 280.384.13-280.fv.1.6 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |