Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $800$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.1.121 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}21&2\\2&35\end{bmatrix}$, $\begin{bmatrix}27&38\\30&13\end{bmatrix}$, $\begin{bmatrix}31&30\\0&37\end{bmatrix}$, $\begin{bmatrix}33&20\\38&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 40.96.1-40.bl.1.1, 40.96.1-40.bl.1.2, 40.96.1-40.bl.1.3, 40.96.1-40.bl.1.4, 80.96.1-40.bl.1.1, 80.96.1-40.bl.1.2, 80.96.1-40.bl.1.3, 80.96.1-40.bl.1.4, 80.96.1-40.bl.1.5, 80.96.1-40.bl.1.6, 80.96.1-40.bl.1.7, 80.96.1-40.bl.1.8, 80.96.1-40.bl.1.9, 80.96.1-40.bl.1.10, 120.96.1-40.bl.1.1, 120.96.1-40.bl.1.2, 120.96.1-40.bl.1.3, 120.96.1-40.bl.1.4, 240.96.1-40.bl.1.1, 240.96.1-40.bl.1.2, 240.96.1-40.bl.1.3, 240.96.1-40.bl.1.4, 240.96.1-40.bl.1.5, 240.96.1-40.bl.1.6, 240.96.1-40.bl.1.7, 240.96.1-40.bl.1.8, 240.96.1-40.bl.1.9, 240.96.1-40.bl.1.10, 280.96.1-40.bl.1.1, 280.96.1-40.bl.1.2, 280.96.1-40.bl.1.3, 280.96.1-40.bl.1.4 |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $15360$ |
Jacobian
| Conductor: | $2^{5}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 800.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + y z $ |
| $=$ | $5 y^{2} + 5 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 10 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{5}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{5^2}\cdot\frac{(25z^{4}-10z^{2}w^{2}+4w^{4})^{3}}{w^{4}z^{4}(5z^{2}-2w^{2})^{2}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.24.0.f.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.a.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.de.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.ee.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.1.a.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.24.1.cy.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.24.1.dy.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.240.17.cj.1 | $40$ | $5$ | $5$ | $17$ | $5$ | $1^{14}\cdot2$ |
| 40.288.17.fn.1 | $40$ | $6$ | $6$ | $17$ | $9$ | $1^{14}\cdot2$ |
| 40.480.33.ln.1 | $40$ | $10$ | $10$ | $33$ | $11$ | $1^{28}\cdot2^{2}$ |
| 80.96.3.bf.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.bf.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.cu.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.cv.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.cw.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.cx.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.dt.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.dt.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.5.cm.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.cn.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bbd.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.9.nz.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.96.3.fj.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fj.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.hk.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.hl.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.hm.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.hn.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.jp.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.jp.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.5.hc.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.hd.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |