Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $320$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ | 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: | 20H3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.3.1540 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&13\\34&29\end{bmatrix}$, $\begin{bmatrix}11&7\\4&39\end{bmatrix}$, $\begin{bmatrix}21&3\\18&21\end{bmatrix}$, $\begin{bmatrix}31&12\\14&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.72.3.ci.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $64$ |
| Full 40-torsion field degree: | $5120$ |
Jacobian
| Conductor: | $2^{16}\cdot5^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 80.2.a.b, 320.2.c.b |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ y u + z t $ |
| $=$ | $ - x u + z w$ | |
| $=$ | $x t + y w$ | |
| $=$ | $6 w^{2} - 4 w t - 2 t^{2} + u^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} y^{4} + 10 x^{4} y^{2} z^{2} + 5 x^{4} z^{4} + 4 x^{2} y^{4} z^{2} - 4 x^{2} y^{2} z^{4} + \cdots + 4 y^{4} z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{8} + 16x^{6} - 88x^{4} + 320x^{2} - 400 $ |
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 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3^6}\cdot\frac{20155392xz^{9}+5205129984xz^{7}u^{2}+7806015360xz^{5}u^{4}+1412277120xz^{3}u^{6}-920018412xzu^{8}+317447424yz^{9}-594584064yz^{7}u^{2}-2017428768yz^{5}u^{4}-688689216yz^{3}u^{6}+184867839yzu^{8}+350720000wt^{8}u+1276432000wt^{6}u^{3}+1538772000wt^{4}u^{5}+533254050wt^{2}u^{7}-277701264wu^{9}+115840000t^{9}u+388704000t^{7}u^{3}+332210000t^{5}u^{5}-14597400t^{3}u^{7}-168031800tu^{9}}{1024xz^{9}-56960xz^{7}u^{2}+5184xz^{5}u^{4}+664xz^{3}u^{6}+444xzu^{8}+16128yz^{9}+7616yz^{7}u^{2}-3600yz^{5}u^{4}+508yz^{3}u^{6}-161yzu^{8}+192000wt^{6}u^{3}-151200wt^{4}u^{5}+35730wt^{2}u^{7}-1744wu^{9}+64000t^{7}u^{3}-74400t^{5}u^{5}+28560t^{3}u^{7}-3680tu^{9}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.72.3.ci.2 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}u$ |
Equation of the image curve:
| $0$ | $=$ | $ 5X^{4}Y^{4}+10X^{4}Y^{2}Z^{2}+4X^{2}Y^{4}Z^{2}+5X^{4}Z^{4}-4X^{2}Y^{2}Z^{4}+4Y^{4}Z^{4}-8X^{2}Z^{6} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 40.72.3.ci.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -\frac{1}{2}u^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 480zwt^{3}u^{3}-96zwtu^{5}+160zt^{4}u^{3}-92zt^{2}u^{5}+\frac{13}{2}zu^{7}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -\frac{3}{2}wu-\frac{1}{2}tu$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.72.1-20.b.1.9 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 40.72.1-20.b.1.14 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.288.5-40.h.2.7 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.z.1.3 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.288.5-40.ce.1.3 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.ch.1.8 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.ej.2.3 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.el.2.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.ew.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.ex.2.8 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.720.19-40.qq.1.7 | $40$ | $5$ | $5$ | $19$ | $1$ | $1^{6}\cdot2^{5}$ |
| 80.288.9-80.u.2.4 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.u.2.8 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.v.2.4 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.v.2.8 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.w.2.6 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.w.2.8 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.x.2.6 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.x.2.8 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.y.2.7 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.y.2.8 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.z.2.6 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.z.2.8 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.ba.2.6 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.ba.2.8 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.bb.2.7 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.bb.2.8 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.5-120.bjs.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bjt.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bkg.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bkh.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bqe.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bqf.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bqs.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bqt.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.15-120.ma.1.28 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 240.288.9-240.bma.2.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bma.2.14 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmb.2.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmb.2.14 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmc.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmc.2.12 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmd.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmd.2.12 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bme.2.7 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bme.2.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmf.2.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmf.2.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmg.2.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmg.2.12 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmh.2.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmh.2.12 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.5-280.xc.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.xd.1.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.xj.1.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.xk.1.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.zg.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.zh.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.zn.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.zo.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |