Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 40.48.1.322 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&11\\16&7\end{bmatrix}$, $\begin{bmatrix}1&25\\10&3\end{bmatrix}$, $\begin{bmatrix}19&38\\6&9\end{bmatrix}$, $\begin{bmatrix}27&4\\8&39\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $15360$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - x z - 5 y^{2} - z^{2} $ |
| $=$ | $2 x^{2} - 7 x z + 5 y^{2} - 7 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 14 x^{2} y^{2} + 15 x^{2} z^{2} + 9 y^{4} - 30 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\cdot3^3\,\frac{442442000000xz^{11}+30660000000xz^{9}w^{2}-294840000xz^{7}w^{4}-22680000xz^{5}w^{6}+340200xz^{3}w^{8}+342827000000z^{12}+58735000000z^{10}w^{2}+1539630000z^{8}w^{4}-61740000z^{6}w^{6}-542700z^{4}w^{8}+24300z^{2}w^{10}-243w^{12}}{110610500000xz^{11}+30078375000xz^{9}w^{2}+3057007500xz^{7}w^{4}+140238000xz^{5}w^{6}+2705400xz^{3}w^{8}+14580xzw^{10}+85706750000z^{12}+32050825000z^{10}w^{2}+4582681875z^{8}w^{4}+311906250z^{6}w^{6}+10035225z^{4}w^{8}+123930z^{2}w^{10}+243w^{12}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.24.1.w.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 20.24.0.i.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.ch.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.dh.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.do.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.1.be.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.bn.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 |
|---|---|---|---|---|---|---|
| 40.96.1.co.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1.co.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.240.17.jy.1 | $40$ | $5$ | $5$ | $17$ | $4$ | $1^{14}\cdot2$ |
| 40.288.17.ye.1 | $40$ | $6$ | $6$ | $17$ | $1$ | $1^{14}\cdot2$ |
| 40.480.33.bps.1 | $40$ | $10$ | $10$ | $33$ | $8$ | $1^{28}\cdot2^{2}$ |
| 120.96.1.qv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1.qv.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.9.ezy.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.9.bpu.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.96.1.py.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1.py.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |