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 | $2^{4}$ | ||
| 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: | 40.48.1.179 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&18\\3&17\end{bmatrix}$, $\begin{bmatrix}23&24\\4&19\end{bmatrix}$, $\begin{bmatrix}27&32\\19&1\end{bmatrix}$, $\begin{bmatrix}37&22\\16&15\end{bmatrix}$ |
| 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: | $2^{5}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 800.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ y z + 2 z w - 2 w^{2} $ |
| $=$ | $5 x^{2} + y^{2} - 2 y z - z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 4 x^{3} z - 5 x^{2} y^{2} + 8 x z^{3} - 4 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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{64y^{12}-768y^{8}w^{4}+3072y^{7}w^{5}-10752y^{6}w^{6}+33792y^{5}w^{7}-96000y^{4}w^{8}+251904y^{3}w^{9}-608256y^{2}w^{10}+1327104yw^{11}-z^{12}+24z^{11}w-264z^{10}w^{2}+1760z^{9}w^{3}-7728z^{8}w^{4}+21504z^{7}w^{5}-24192z^{6}w^{6}-89856z^{5}w^{7}+561600z^{4}w^{8}-1500160z^{3}w^{9}+1883136z^{2}w^{10}+1634304zw^{11}-2480128w^{12}}{w^{4}(16y^{4}w^{4}-128y^{3}w^{5}+704y^{2}w^{6}-3200yw^{7}+z^{8}-20z^{7}w+182z^{6}w^{2}-996z^{5}w^{3}+3649z^{4}w^{4}-9424z^{3}w^{5}+17584z^{2}w^{6}-23872zw^{7}+12896w^{8})}$ |
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.bf.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 20.24.0.j.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.ch.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.eo.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.1.da.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.24.1.dj.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.24.1.em.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 |
|---|---|---|---|---|---|---|
| 40.96.1.cs.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1.cs.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.240.17.xx.1 | $40$ | $5$ | $5$ | $17$ | $5$ | $1^{14}\cdot2$ |
| 40.288.17.cev.1 | $40$ | $6$ | $6$ | $17$ | $4$ | $1^{14}\cdot2$ |
| 40.480.33.dwx.1 | $40$ | $10$ | $10$ | $33$ | $7$ | $1^{28}\cdot2^{2}$ |
| 120.96.1.tu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1.tu.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.9.bexl.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.9.dnu.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.96.1.qc.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1.qc.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |