Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $320$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.1.15 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&11\\6&5\end{bmatrix}$, $\begin{bmatrix}7&32\\34&25\end{bmatrix}$, $\begin{bmatrix}9&17\\18&23\end{bmatrix}$, $\begin{bmatrix}37&4\\6&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.72.1.t.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $32$ |
| Full 40-torsion field degree: | $5120$ |
Jacobian
| Conductor: | $2^{6}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 320.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 5x - 5 $ |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{10}}\cdot\frac{48x^{2}y^{22}+113536x^{2}y^{20}z^{2}+18764800x^{2}y^{18}z^{4}+951164928x^{2}y^{16}z^{6}+23479582720x^{2}y^{14}z^{8}+358406422528x^{2}y^{12}z^{10}+3850345906176x^{2}y^{10}z^{12}+30161743380480x^{2}y^{8}z^{14}+174225079926784x^{2}y^{6}z^{16}+733311978700800x^{2}y^{4}z^{18}+2048132464508928x^{2}y^{2}z^{20}+3429514205986816x^{2}z^{22}+1056xy^{22}z+850688xy^{20}z^{3}+97290240xy^{18}z^{5}+4167745536xy^{16}z^{7}+94294507520xy^{14}z^{9}+1370033750016xy^{12}z^{11}+14199916855296xy^{10}z^{13}+108074127851520xy^{8}z^{15}+609857975615488xy^{6}z^{17}+2505920143687680xy^{4}z^{19}+6864972646711296xy^{2}z^{21}+11098195492864000xz^{23}+y^{24}+13264y^{22}z^{2}+4152320y^{20}z^{4}+290933760y^{18}z^{6}+9013301248y^{16}z^{8}+164075012096y^{14}z^{10}+2052266655744y^{12}z^{12}+18749168025600y^{10}z^{14}+127597354680320y^{8}z^{16}+650056453586944y^{6}z^{18}+2382959524970496y^{4}z^{20}+5876735031640064y^{2}z^{22}+7668750006353920z^{24}}{z^{6}y^{4}(y^{2}+8z^{2})^{2}(x^{2}y^{8}+1184x^{2}y^{6}z^{2}+69248x^{2}y^{4}z^{4}+1067008x^{2}y^{2}z^{6}+4771840x^{2}z^{8}+22xy^{8}z+6688xy^{6}z^{3}+285696xy^{4}z^{5}+3782656xy^{2}z^{7}+15441920xz^{9}+201y^{8}z^{2}+21696y^{6}z^{4}+514688y^{4}z^{6}+4190208y^{2}z^{8}+10670080z^{10})}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 5.24.0-5.a.1.1 | $5$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 8.6.0.d.1 | $8$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.72.0-10.a.2.3 | $10$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.72.0-10.a.2.2 | $40$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.288.5-40.d.1.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.bs.1.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.288.5-40.dn.1.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.288.5-40.ds.1.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.fq.1.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.fu.1.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.gb.1.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.288.5-40.gd.1.1 | $40$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 40.720.13-40.bf.1.1 | $40$ | $5$ | $5$ | $13$ | $3$ | $1^{6}\cdot2^{3}$ |
| 120.288.5-120.bdf.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bdh.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bdt.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bdv.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bpf.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bph.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bpt.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bpv.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.13-120.dz.2.5 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 280.288.5-280.qh.1.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.qi.1.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.qo.1.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.qp.1.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.sl.1.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.sm.1.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.ss.1.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.st.1.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |