Invariants
| Level: | $40$ | $\SL_2$-level: | $10$ | Newform level: | $1600$ | ||
| Index: | $30$ | $\PSL_2$-index: | $30$ | ||||
| Genus: | $1 = 1 + \frac{ 30 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $10^{3}$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 10E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.30.1.3 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}8&23\\9&2\end{bmatrix}$, $\begin{bmatrix}17&5\\31&38\end{bmatrix}$, $\begin{bmatrix}18&23\\5&12\end{bmatrix}$, $\begin{bmatrix}21&17\\3&4\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $24576$ |
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} - 133x + 363 $ |
Rational points
This modular curve has infinitely many rational points, including 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 30 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^5\cdot5^5}\cdot\frac{50x^{2}y^{8}-600000x^{2}y^{6}z^{2}+18780000000x^{2}y^{4}z^{4}+897160000000000x^{2}y^{2}z^{6}+763970000000000000x^{2}z^{8}+1200xy^{8}z-75200000xy^{6}z^{3}+930720000000xy^{4}z^{5}+16261440000000000xy^{2}z^{7}+18680880000000000000xz^{9}-y^{10}-24050y^{8}z^{2}-1015000000y^{6}z^{4}+45738820000000y^{4}z^{6}+152028240000000000y^{2}z^{8}-61190370000000000000z^{10}}{z^{3}(100x^{2}y^{4}z-300000x^{2}y^{2}z^{3}-100000000x^{2}z^{5}-xy^{6}+2400xy^{4}z^{2}-200000xy^{2}z^{4}-400000000xz^{6}-17y^{6}z-8100y^{4}z^{3}+43300000y^{2}z^{5}+12100000000z^{7})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.60.3.t.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.60.3.u.1 | $40$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 40.60.3.w.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.60.3.x.1 | $40$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 40.60.3.bf.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.60.3.bg.1 | $40$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 40.60.3.bi.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.60.3.bj.1 | $40$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 40.90.3.c.1 | $40$ | $3$ | $3$ | $3$ | $2$ | $1^{2}$ |
| 40.120.6.a.1 | $40$ | $4$ | $4$ | $6$ | $3$ | $1^{5}$ |
| 120.60.3.ca.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.cb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.cg.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.ch.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.dz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.ea.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.ec.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.ed.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.90.4.be.1 | $120$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 120.120.8.jm.1 | $120$ | $4$ | $4$ | $8$ | $?$ | not computed |
| 200.150.9.b.1 | $200$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 280.60.3.ea.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.60.3.eb.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.60.3.eg.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.60.3.eh.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.60.3.em.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.60.3.en.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.60.3.es.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.60.3.et.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.240.18.bw.1 | $280$ | $8$ | $8$ | $18$ | $?$ | not computed |