Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $20$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $2^{6}\cdot10^{6}$ | Cusp orbits | $2^{6}$ | ||
| 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: | 10K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.1.520 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&30\\30&11\end{bmatrix}$, $\begin{bmatrix}17&10\\0&27\end{bmatrix}$, $\begin{bmatrix}19&0\\32&27\end{bmatrix}$, $\begin{bmatrix}37&4\\20&1\end{bmatrix}$, $\begin{bmatrix}37&32\\6&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.72.1.a.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: | $2^{2}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 20.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 5 x^{2} - 3 z^{2} - 4 z w $ |
| $=$ | $5 x y + 5 y^{2} + 2 z^{2} + 3 z w + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} + 6 x^{2} z^{2} + 4 x y z^{2} + y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(269z^{6}+172z^{5}w-1760z^{4}w^{2}-3840z^{3}w^{3}-3040z^{2}w^{4}-848zw^{5}+16w^{6})^{3}}{z(z+w)^{2}(2z+w)^{10}(3z+4w)^{5}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 20.72.1.a.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ 5X^{4}+6X^{2}Z^{2}+4XYZ^{2}+Y^{2}Z^{2}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.72.1-10.a.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.1-10.a.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.288.5-20.c.2.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-20.c.2.7 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-20.d.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-20.d.2.10 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-20.d.2.17 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-20.g.2.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-20.g.2.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-20.g.2.3 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.g.2.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.g.2.3 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.g.2.5 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-20.h.2.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-20.h.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-20.h.2.3 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.j.2.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.j.2.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.j.2.7 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.s.2.2 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.288.5-40.s.2.3 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.288.5-40.s.2.5 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.288.5-40.v.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.v.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.v.2.7 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.720.13-20.a.1.2 | $40$ | $5$ | $5$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
| 120.288.5-60.ba.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.ba.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.ba.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bb.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bb.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bb.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bm.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bm.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bm.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bn.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bn.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bn.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.da.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.da.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.da.2.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dd.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dd.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dd.2.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ek.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ek.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ek.2.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.en.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.en.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.en.2.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.13-60.a.1.6 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 280.288.5-140.c.2.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.c.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.c.2.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.d.2.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.d.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.d.2.8 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.g.2.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.g.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.g.2.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.g.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.g.2.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.g.2.9 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.h.2.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.h.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.h.2.8 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.j.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.j.2.8 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.j.2.14 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.s.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.s.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.s.2.9 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.v.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.v.2.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.v.2.14 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |