Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $80$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 40.144.1.389 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&8\\4&7\end{bmatrix}$, $\begin{bmatrix}5&34\\24&5\end{bmatrix}$, $\begin{bmatrix}23&33\\6&35\end{bmatrix}$, $\begin{bmatrix}33&38\\36&5\end{bmatrix}$, $\begin{bmatrix}37&2\\30&39\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.72.1.d.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: | $2^{4}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 80.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{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.
| Weierstrass model |
|---|
| $(0:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{24x^{2}y^{22}+7096x^{2}y^{20}z^{2}+146600x^{2}y^{18}z^{4}+928872x^{2}y^{16}z^{6}+2866160x^{2}y^{14}z^{8}+5468848x^{2}y^{12}z^{10}+7343952x^{2}y^{10}z^{12}+7191120x^{2}y^{8}z^{14}+5192312x^{2}y^{6}z^{16}+2731800x^{2}y^{4}z^{18}+953736x^{2}y^{2}z^{20}+199624x^{2}z^{22}+240xy^{22}z+19488xy^{20}z^{3}+233440xy^{18}z^{5}+1106160xy^{16}z^{7}+2889120xy^{14}z^{9}+4983680xy^{12}z^{11}+6198144xy^{10}z^{13}+5692320xy^{8}z^{15}+3895280xy^{6}z^{17}+1935840xy^{4}z^{19}+644640xy^{2}z^{21}+123376xz^{23}+y^{24}+1532y^{22}z^{2}+53362y^{20}z^{4}+414860y^{18}z^{6}+1415215y^{16}z^{8}+2846072y^{14}z^{10}+3969724y^{12}z^{12}+4005240y^{10}z^{14}+2961455y^{8}z^{16}+1597580y^{6}z^{18}+568434y^{4}z^{20}+123388y^{2}z^{22}+z^{24}}{z^{6}y^{4}(y^{2}+z^{2})^{2}(x^{2}y^{8}+148x^{2}y^{6}z^{2}+1082x^{2}y^{4}z^{4}+2084x^{2}y^{2}z^{6}+1165x^{2}z^{8}+10xy^{8}z+270xy^{6}z^{3}+1150xy^{4}z^{5}+1610xy^{2}z^{7}+720xz^{9}+45y^{8}z^{2}+506y^{6}z^{4}+1165y^{4}z^{6}+720y^{2}z^{8})}$ |
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 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.1-20.b.1.16 | $40$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.288.3-40.g.1.10 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.288.3-40.g.1.14 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.288.3-40.h.1.13 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.288.3-40.h.1.15 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.288.3-40.k.2.4 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.288.3-40.k.2.12 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.288.3-40.l.2.10 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.288.3-40.l.2.14 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.288.5-20.a.1.10 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-20.i.1.6 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-20.q.1.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-20.s.1.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.y.1.7 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.288.5-40.ch.1.8 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.ek.1.8 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.ex.1.8 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.ff.2.11 | $40$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 40.288.7-40.ff.2.12 | $40$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 40.288.7-40.fg.2.7 | $40$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 40.288.7-40.fg.2.8 | $40$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 40.288.7-40.fj.1.13 | $40$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 40.288.7-40.fj.1.14 | $40$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 40.288.7-40.fk.1.11 | $40$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 40.288.7-40.fk.1.12 | $40$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 40.720.13-20.g.1.3 | $40$ | $5$ | $5$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 120.288.3-120.n.1.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.n.1.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.o.1.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.o.1.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.r.2.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.r.2.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.s.2.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.s.2.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.5-60.ea.1.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.ec.1.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fo.1.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.fq.1.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bcb.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bcp.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bob.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bop.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.7-120.fpv.2.20 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.fpv.2.24 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.fpw.2.20 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.fpw.2.24 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.fpz.1.26 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.fpz.1.28 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.fqa.1.26 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.fqa.1.28 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.432.13-60.y.2.15 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 280.288.3-280.g.1.18 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.288.3-280.g.1.22 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.288.3-280.h.1.25 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.288.3-280.h.1.27 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.288.3-280.k.2.6 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.288.3-280.k.2.14 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.288.3-280.l.2.19 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.288.3-280.l.2.23 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.288.5-140.ce.1.12 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.cf.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.cm.1.12 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.cn.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.pc.1.13 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.pj.1.14 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.rg.1.14 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.rn.1.14 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.7-280.gt.2.21 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.gt.2.23 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.gu.2.10 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.gu.2.14 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.gx.1.25 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.gx.1.27 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.gy.1.18 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.gy.1.22 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |