Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $1600$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 40.48.1.182 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&16\\39&25\end{bmatrix}$, $\begin{bmatrix}9&16\\1&35\end{bmatrix}$, $\begin{bmatrix}25&32\\16&13\end{bmatrix}$, $\begin{bmatrix}29&6\\9&15\end{bmatrix}$ |
| 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: | $2^{6}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1600.2.a.n |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ y z - 2 z w - 2 w^{2} $ |
| $=$ | $10 x^{2} - y^{2} + 2 y z + z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 4 x^{3} z + 10 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 |
| 40.24.0.cj.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.cy.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.em.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.1.cw.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.dd.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.en.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.ct.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1.ct.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.240.17.yb.1 | $40$ | $5$ | $5$ | $17$ | $7$ | $1^{14}\cdot2$ |
| 40.288.17.cez.1 | $40$ | $6$ | $6$ | $17$ | $4$ | $1^{14}\cdot2$ |
| 40.480.33.dxb.1 | $40$ | $10$ | $10$ | $33$ | $12$ | $1^{28}\cdot2^{2}$ |
| 120.96.1.tv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1.tv.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.9.bexp.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.9.dny.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.96.1.qd.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1.qd.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |