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: | 8G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.1.815 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&20\\28&9\end{bmatrix}$, $\begin{bmatrix}19&20\\18&37\end{bmatrix}$, $\begin{bmatrix}23&28\\18&31\end{bmatrix}$, $\begin{bmatrix}31&0\\8&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.1.x.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| 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 $ | $=$ | $ 5 x y + 5 y^{2} + w^{2} $ |
| $=$ | $5 x^{2} - 5 x y - 2 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} - 10 x^{2} y^{2} + 15 x^{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 to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^2\,\frac{10080y^{2}z^{10}-15120y^{2}z^{8}w^{2}+720y^{2}z^{6}w^{4}+360y^{2}z^{4}w^{6}-1890y^{2}z^{2}w^{8}+315y^{2}w^{10}-2048z^{12}+6144z^{10}w^{2}-4080z^{8}w^{4}+512z^{6}w^{6}-192z^{4}w^{8}-120z^{2}w^{10}+31w^{12}}{w^{4}z^{4}(10y^{2}z^{2}+5y^{2}w^{2}+w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.48.1.x.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{5}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}-10X^{2}Y^{2}+15X^{2}Z^{2}+25Z^{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.1.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.0-40.h.2.8 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-40.h.2.31 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-40.i.2.10 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-40.i.2.25 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.1-8.d.1.4 | $40$ | $2$ | $2$ | $1$ | $0$ | 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.e.2.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.p.2.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.ba.2.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.bg.2.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.bi.2.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.bo.2.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.bq.2.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.br.2.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.480.17-40.bl.1.14 | $40$ | $5$ | $5$ | $17$ | $2$ | $1^{6}\cdot2^{5}$ |
| 40.576.17-40.ct.2.24 | $40$ | $6$ | $6$ | $17$ | $0$ | $1^{6}\cdot2\cdot4^{2}$ |
| 40.960.33-40.gf.1.30 | $40$ | $10$ | $10$ | $33$ | $4$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 120.192.1-120.de.1.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.dg.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.du.2.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ee.2.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ek.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.eu.2.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.fi.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.fk.1.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.288.9-120.ep.2.47 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.384.9-120.cv.1.55 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.192.1-280.dm.1.12 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.do.1.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.dw.1.11 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.ec.1.8 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.em.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.es.1.9 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.fa.1.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.fc.1.11 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |