Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot4\cdot5^{2}\cdot20$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 20D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.36.1.10 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}23&10\\6&37\end{bmatrix}$, $\begin{bmatrix}25&14\\3&11\end{bmatrix}$, $\begin{bmatrix}27&14\\33&33\end{bmatrix}$, $\begin{bmatrix}31&8\\32&37\end{bmatrix}$, $\begin{bmatrix}37&32\\31&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $64$ |
| Full 40-torsion field degree: | $20480$ |
Jacobian
| Conductor: | $2^{6}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1600.2.a.w |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} + 367x - 2863 $ |
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.
| Weierstrass model |
|---|
| $(0:1:0)$, $(7:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{10}\cdot5^{10}}\cdot\frac{180x^{2}y^{10}-6900000x^{2}y^{8}z^{2}+142800000000x^{2}y^{6}z^{4}+391140000000000x^{2}y^{4}z^{6}-2211580000000000000x^{2}y^{2}z^{8}-2211590000000000000000x^{2}z^{10}+10980xy^{10}z-246600000xy^{8}z^{3}-1754700000000xy^{6}z^{5}+19120440000000000xy^{4}z^{7}+42027020000000000000xy^{2}z^{9}-24328740000000000000000xz^{11}+y^{12}+214320y^{10}z^{2}+2730300000y^{8}z^{4}-34034300000000y^{6}z^{6}-174925940000000000y^{4}z^{8}-75089720000000000000y^{2}z^{10}+278794090000000000000000z^{12}}{z^{6}(x^{2}y^{4}-32000x^{2}y^{2}z^{2}-128000000x^{2}z^{4}+46xy^{4}z+608000xy^{2}z^{3}-1408000000xz^{5}-671y^{4}z^{2}+3712000y^{2}z^{4}+16128000000z^{6})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(10)$ | $10$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.6.0.d.1 | $40$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.72.1.br.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.1.br.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.1.bs.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.1.bs.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.1.bu.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.1.bu.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.1.bv.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.1.bv.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.3.d.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.72.3.o.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.72.3.be.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.72.3.bg.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.72.3.dn.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.72.3.dn.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.72.3.do.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.72.3.do.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.72.3.dq.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.72.3.dq.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.72.3.dr.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.72.3.dr.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.72.3.ds.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.72.3.du.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.72.3.eb.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.72.3.ed.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.180.7.cp.1 | $40$ | $5$ | $5$ | $7$ | $2$ | $1^{6}$ |
| 120.72.1.hf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.hf.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.hg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.hg.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.hi.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.hi.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.hj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.hj.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.3.cyl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cyn.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cyr.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cyt.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edl.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edm.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edo.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edo.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edp.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edp.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ege.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.egf.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.egk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.egl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.108.7.l.1 | $120$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 120.144.7.hmo.1 | $120$ | $4$ | $4$ | $7$ | $?$ | not computed |
| 200.180.7.h.1 | $200$ | $5$ | $5$ | $7$ | $?$ | not computed |
| 280.72.1.bv.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bv.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bw.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bw.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.by.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.by.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bz.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bz.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.3.dz.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.ea.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.ec.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.ed.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.ex.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.ex.2 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.ey.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.ey.2 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.fa.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.fa.2 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.fb.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.fb.2 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.fv.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.fw.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.fy.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.fz.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.288.19.z.1 | $280$ | $8$ | $8$ | $19$ | $?$ | not computed |