Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $800$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.1.69 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&26\\27&5\end{bmatrix}$, $\begin{bmatrix}11&38\\10&21\end{bmatrix}$, $\begin{bmatrix}25&2\\11&15\end{bmatrix}$, $\begin{bmatrix}31&28\\20&3\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $30720$ |
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 $ | $=$ | $ 4 x^{2} + z w $ |
| $=$ | $10 y^{2} - 4 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 10 y^{2} z^{2} - 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^8\,\frac{(z-w)^{3}(z+w)^{3}}{w^{2}z^{4}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.12.0.z.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.12.0.bk.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.12.1.h.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 |
|---|---|---|---|---|---|---|
| 40.48.1.hu.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.hv.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.hw.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.hx.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.ji.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.jj.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.jk.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.jl.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.120.9.fb.1 | $40$ | $5$ | $5$ | $9$ | $5$ | $1^{6}\cdot2$ |
| 40.144.9.kh.1 | $40$ | $6$ | $6$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 40.240.17.baj.1 | $40$ | $10$ | $10$ | $17$ | $9$ | $1^{12}\cdot2^{2}$ |
| 80.48.3.bu.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.bu.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.eq.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.eq.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.es.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.es.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.gy.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.gy.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.1.cla.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.clb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.clc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cld.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cmg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cmh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cmi.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cmj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.chb.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.5.vr.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 240.48.3.gi.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.gi.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.km.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.km.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.ko.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.ko.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.si.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.si.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.48.1.bsk.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bsl.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bsm.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bsn.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bta.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.btb.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.btc.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.btd.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.13.lz.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |