Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.1.103 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}19&24\\31&13\end{bmatrix}$, $\begin{bmatrix}19&34\\39&1\end{bmatrix}$, $\begin{bmatrix}23&26\\33&37\end{bmatrix}$, $\begin{bmatrix}25&12\\29&7\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 80.48.1-40.y.1.1, 80.48.1-40.y.1.2, 80.48.1-40.y.1.3, 80.48.1-40.y.1.4, 80.48.1-40.y.1.5, 80.48.1-40.y.1.6, 80.48.1-40.y.1.7, 80.48.1-40.y.1.8, 240.48.1-40.y.1.1, 240.48.1-40.y.1.2, 240.48.1-40.y.1.3, 240.48.1-40.y.1.4, 240.48.1-40.y.1.5, 240.48.1-40.y.1.6, 240.48.1-40.y.1.7, 240.48.1-40.y.1.8 |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $30720$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ y^{2} + y z - z^{2} + w^{2} $ |
| $=$ | $20 x^{2} + y w - 2 z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 5 x^{2} y z + 5 y^{2} z^{2} + 25 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{10}w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{1000yz^{5}-575yz^{3}w^{2}+60yzw^{4}-625z^{6}+800z^{4}w^{2}-210z^{2}w^{4}+8w^{6}}{w^{4}(5yz-5z^{2}+w^{2})}$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.1.c.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 20.12.0.k.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.12.0.cb.1 | $40$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.48.1.eu.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ev.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ew.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ex.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.gc.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.gd.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ge.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.gf.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.bo.1 | $40$ | $5$ | $5$ | $9$ | $2$ | $1^{6}\cdot2$ |
| 40.144.9.cm.1 | $40$ | $6$ | $6$ | $9$ | $2$ | $1^{6}\cdot2$ |
| 40.240.17.lu.1 | $40$ | $10$ | $10$ | $17$ | $3$ | $1^{12}\cdot2^{2}$ |
| 120.48.1.rm.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.rn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.ro.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.rp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.ss.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.st.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.su.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.sv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.cu.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.5.by.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.oc.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.od.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.oe.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.of.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.os.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ot.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ou.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ov.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.13.by.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |