Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $1600$ | ||
| 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: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.1.60 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&10\\22&23\end{bmatrix}$, $\begin{bmatrix}35&22\\32&33\end{bmatrix}$, $\begin{bmatrix}35&34\\2&17\end{bmatrix}$, $\begin{bmatrix}37&36\\37&23\end{bmatrix}$ |
| 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: | $30720$ |
Jacobian
| Conductor: | $2^{6}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1600.2.a.n |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x z - y z - 2 y w $ |
| $=$ | $8 x^{2} - 5 y^{2} - 2 z^{2} + 2 z w + 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} - 2 x^{3} z + 3 x^{2} y^{2} - 2 x^{2} z^{2} - 8 x y^{2} z - 8 y^{2} z^{2} $ |
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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^4\,\frac{234000xy^{3}w^{2}+1156000xyw^{4}+10125y^{6}+204750y^{4}w^{2}+939500y^{2}w^{4}+3192z^{6}+13464z^{5}w-66840z^{4}w^{2}-187960z^{3}w^{3}+384640z^{2}w^{4}+290944zw^{5}+648w^{6}}{54000xy^{3}w^{2}-836000xyw^{4}+30375y^{6}+47250y^{4}w^{2}-659500y^{2}w^{4}+6696z^{6}+17352z^{5}w+30360z^{4}w^{2}+119960z^{3}w^{3}-268160z^{2}w^{4}-203168zw^{5}+1944w^{6}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.0.q.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 20.12.0.o.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.12.1.g.1 | $40$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.48.1.bp.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.cc.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.cv.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.da.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.hn.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.hr.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ic.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ig.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.dh.1 | $40$ | $5$ | $5$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 40.144.9.hl.1 | $40$ | $6$ | $6$ | $9$ | $1$ | $1^{6}\cdot2$ |
| 40.240.17.sx.1 | $40$ | $10$ | $10$ | $17$ | $4$ | $1^{12}\cdot2^{2}$ |
| 120.48.1.bbb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bbj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bbr.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bbz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cfj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cfr.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cfy.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cgg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.bof.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.5.pd.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.bcx.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bdb.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bdn.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bdr.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bmt.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bmx.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bnj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bnn.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.13.hx.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |