Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $80$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20H3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.3.979 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&13\\32&5\end{bmatrix}$, $\begin{bmatrix}21&7\\20&33\end{bmatrix}$, $\begin{bmatrix}23&36\\16&23\end{bmatrix}$, $\begin{bmatrix}35&17\\34&23\end{bmatrix}$, $\begin{bmatrix}37&2\\16&33\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.72.3.r.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: | $2^{12}\cdot5^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 80.2.a.b, 80.2.c.a |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ - x t^{2} + z w t - z t^{2} $ |
| $=$ | $ - x w t + z w^{2} - z w t$ | |
| $=$ | $x w t + y w t - z t^{2}$ | |
| $=$ | $x w^{2} + y w^{2} - z w t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{5} y^{2} + x^{5} z^{2} - 12 x^{4} y^{2} z - x^{4} z^{3} + 12 x^{3} y^{2} z^{2} - x^{3} z^{4} + \cdots - y^{2} z^{5} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{7} - 7x^{6} + 23x^{5} - 38x^{4} + 23x^{3} - 7x^{2} + x $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:1:1:0:0)$, $(0:0:0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{9351861xyz^{9}-8972201xyz^{7}t^{2}+595066xyz^{5}t^{4}+190965xyz^{3}t^{6}-14204xyzt^{8}-38831741xz^{10}+2142220xz^{8}t^{2}+2385830xz^{6}t^{4}-237145xz^{4}t^{6}+4870xz^{2}t^{8}+3xt^{10}+9220633y^{2}z^{9}-4717340y^{2}z^{7}t^{2}+583563y^{2}z^{5}t^{4}-17002y^{2}z^{3}t^{6}-149y^{2}zt^{8}-32617853yz^{10}+6258166yz^{8}t^{2}-188281yz^{6}t^{4}+111630yz^{4}t^{6}-9931yz^{2}t^{8}-100yw^{9}t+200yw^{8}t^{2}+100yw^{7}t^{3}-200yw^{6}t^{4}+25yw^{5}t^{5}-250yw^{3}t^{7}-250yw^{2}t^{8}+198ywt^{9}-25yt^{10}+23397245z^{11}-414095z^{9}t^{2}-1666990z^{7}t^{4}+156930z^{5}t^{6}-2995z^{3}t^{8}+5zt^{10}}{z^{3}(5412xyz^{6}-983xyz^{4}t^{2}-226xyz^{2}t^{4}-5xyt^{6}-22472xz^{7}-3227xz^{5}t^{2}-85xz^{3}t^{4}-3xzt^{6}+5336y^{2}z^{6}-210y^{2}z^{4}t^{2}-8y^{2}z^{2}t^{4}-18876yz^{7}-1250yz^{5}t^{2}-189yz^{3}t^{4}-5yzt^{6}+13540z^{8}+2112z^{6}t^{2}+55z^{4}t^{4}+2z^{2}t^{6})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 20.72.3.r.2 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{5}Y^{2}-12X^{4}Y^{2}Z+X^{5}Z^{2}+12X^{3}Y^{2}Z^{2}-X^{4}Z^{3}-4X^{2}Y^{2}Z^{3}-X^{3}Z^{4}+XY^{2}Z^{4}-Y^{2}Z^{5} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 20.72.3.r.2 :
| $\displaystyle X$ | $=$ | $\displaystyle w^{2}t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4zw^{8}t^{3}-12zw^{7}t^{4}+12zw^{6}t^{5}-4zw^{5}t^{6}+zw^{4}t^{7}-zw^{3}t^{8}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w^{2}t-wt^{2}$ |
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.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 40.72.1-20.b.1.9 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.288.5-20.a.1.10 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-20.j.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-20.q.2.6 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-20.t.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.bd.2.3 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.288.5-40.cp.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.en.2.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.ff.2.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.9-40.fw.1.13 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{3}$ |
| 40.288.9-40.fw.1.14 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{3}$ |
| 40.288.9-40.fx.1.10 | $40$ | $2$ | $2$ | $9$ | $2$ | $2^{3}$ |
| 40.288.9-40.fx.1.12 | $40$ | $2$ | $2$ | $9$ | $2$ | $2^{3}$ |
| 40.288.9-40.fy.2.6 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{3}$ |
| 40.288.9-40.fy.2.14 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{3}$ |
| 40.288.9-40.fz.2.4 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{3}$ |
| 40.288.9-40.fz.2.12 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{3}$ |
| 40.288.9-40.ga.2.4 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{3}$ |
| 40.288.9-40.ga.2.8 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{3}$ |
| 40.288.9-40.gb.2.10 | $40$ | $2$ | $2$ | $9$ | $4$ | $2^{3}$ |
| 40.288.9-40.gb.2.12 | $40$ | $2$ | $2$ | $9$ | $4$ | $2^{3}$ |
| 40.288.9-40.gc.1.6 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{3}$ |
| 40.288.9-40.gc.1.14 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{3}$ |
| 40.288.9-40.gd.1.7 | $40$ | $2$ | $2$ | $9$ | $2$ | $2^{3}$ |
| 40.288.9-40.gd.1.15 | $40$ | $2$ | $2$ | $9$ | $2$ | $2^{3}$ |
| 40.720.19-20.cf.1.9 | $40$ | $5$ | $5$ | $19$ | $1$ | $1^{6}\cdot2^{5}$ |
| 120.288.5-60.ez.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fb.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fx.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fz.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bkb.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bkp.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bqn.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.brb.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.9-120.iou.1.4 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.iou.1.8 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.iov.1.6 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.iov.1.8 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.iow.1.10 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.iow.1.14 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.iox.1.10 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.iox.1.14 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.ioy.1.11 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.ioy.1.12 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.ioz.1.10 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.ioz.1.12 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.ipa.1.4 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.ipa.1.12 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.ipb.1.4 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.ipb.1.12 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.432.15-60.dl.1.41 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 280.288.5-140.cw.1.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.cx.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.de.2.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.df.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.xs.2.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.xz.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.zw.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bad.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.9-280.gm.1.13 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.gm.1.15 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.gn.1.10 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.gn.1.14 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.go.1.11 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.go.1.15 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.gp.1.6 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.gp.1.14 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.gq.1.6 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.gq.1.14 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.gr.1.11 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.gr.1.15 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.gs.1.10 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.gs.1.14 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.gt.1.13 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.9-280.gt.1.15 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |