Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $96$ | $\PSL_2$-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: | 40.96.1.898 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&4\\22&13\end{bmatrix}$, $\begin{bmatrix}5&16\\8&15\end{bmatrix}$, $\begin{bmatrix}23&0\\30&11\end{bmatrix}$, $\begin{bmatrix}31&36\\26&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.1.u.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $7680$ |
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 $ | $=$ | $ 3 x^{2} - x y + 2 x z - 2 y^{2} - 2 y z + 2 z^{2} $ |
| $=$ | $10 x y + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 25 x^{4} + 5 x^{3} y - x^{2} y^{2} - 5 x^{2} z^{2} + 2 x y z^{2} - 6 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 to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^4}{3^8\cdot5}\cdot\frac{630784000000xz^{11}-46802136000000xz^{9}w^{2}-11618346240000xz^{7}w^{4}-1134083484000xz^{5}w^{6}-45930823200xz^{3}w^{8}-652695570xzw^{10}-504561760000000y^{2}z^{10}-140563080000000y^{2}z^{8}w^{2}-16274048400000y^{2}z^{6}w^{4}-856302300000y^{2}z^{4}w^{6}-19177560000y^{2}z^{2}w^{8}-87637950y^{2}w^{10}+311731616000000yz^{11}+111752616000000yz^{9}w^{2}+17400223440000yz^{7}w^{4}+1339110684000yz^{5}w^{6}+48681826200yz^{3}w^{8}+652695570yzw^{10}+47104000000z^{12}-30854492800000z^{10}w^{2}-10644621120000z^{8}w^{4}-1381610880000z^{6}w^{6}-80232130800z^{4}w^{8}-1882700820z^{2}w^{10}-14432499w^{12}}{w^{8}(640xz^{3}+138xzw^{2}+2000y^{2}z^{2}+30y^{2}w^{2}-1240yz^{3}-138yzw^{2}+160z^{4}+188z^{2}w^{2}+3w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.48.1.u.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 5z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 25X^{4}+5X^{3}Y-X^{2}Y^{2}-5X^{2}Z^{2}+2XYZ^{2}-6Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.1-8.d.1.9 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.0-20.c.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-20.c.1.14 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-40.l.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-40.l.1.14 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.1-8.d.1.6 | $40$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.192.1-40.ba.1.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.ba.2.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.bc.1.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.bc.2.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.be.1.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.be.2.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.bg.1.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.bg.2.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.480.17-40.bh.1.15 | $40$ | $5$ | $5$ | $17$ | $5$ | $1^{14}\cdot2$ |
| 40.576.17-40.ck.1.30 | $40$ | $6$ | $6$ | $17$ | $3$ | $1^{14}\cdot2$ |
| 40.960.33-40.fr.1.31 | $40$ | $10$ | $10$ | $33$ | $9$ | $1^{28}\cdot2^{2}$ |
| 120.192.1-120.dk.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.dk.2.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.dm.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.dm.2.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.do.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.do.2.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.dq.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.dq.2.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.288.9-120.dx.1.57 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.384.9-120.ci.1.60 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.192.1-280.dc.1.14 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.dc.2.16 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.de.1.15 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.de.2.16 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.dg.1.16 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.dg.2.12 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.di.1.16 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.di.2.12 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |