Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $80$ | ||
| Index: | $72$ | $\PSL_2$-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: | 40.72.1.160 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&39\\10&27\end{bmatrix}$, $\begin{bmatrix}5&14\\24&25\end{bmatrix}$, $\begin{bmatrix}7&16\\4&19\end{bmatrix}$, $\begin{bmatrix}27&16\\38&5\end{bmatrix}$, $\begin{bmatrix}29&38\\26&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.36.1.b.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $64$ |
| Full 40-torsion field degree: | $10240$ |
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} + 4x - 4 $ |
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 |
|---|
| $(1:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{18x^{2}y^{10}+231870x^{2}y^{8}z^{2}+160411560x^{2}y^{6}z^{4}+2836033434x^{2}y^{4}z^{6}+5360733242x^{2}y^{2}z^{8}+5211922921x^{2}z^{10}+819xy^{10}z+2126268xy^{8}z^{3}+512947965xy^{6}z^{5}-122871624xy^{4}z^{7}+6960980885xy^{2}z^{9}-7562391552xz^{11}+y^{12}+10983y^{10}z^{2}+23771328y^{8}z^{4}+1733869315y^{6}z^{6}+10433358193y^{4}z^{8}+23727088765y^{2}z^{10}+32868046756z^{12}}{z(50x^{2}y^{8}z+2918x^{2}y^{6}z^{3}-104248x^{2}y^{4}z^{5}-958112x^{2}y^{2}z^{7}-515968x^{2}z^{9}+xy^{10}-320xy^{8}z^{2}+15859xy^{6}z^{4}+333620xy^{4}z^{6}+39632xy^{2}z^{8}-4376384xz^{10}+15y^{10}z-1205y^{8}z^{3}-33613y^{6}z^{5}+80052y^{4}z^{7}+2000144y^{2}z^{9}+4892352z^{11})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.12.0-4.b.1.3 | $40$ | $6$ | $6$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.144.1-20.d.1.7 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.1-20.d.2.9 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.1-20.e.1.5 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.1-20.e.2.7 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.3-20.a.1.11 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-20.g.1.3 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-20.m.1.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.144.3-20.n.1.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-20.q.1.7 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-20.q.2.8 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-20.r.1.7 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-20.r.2.8 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.360.7-20.g.1.9 | $40$ | $5$ | $5$ | $7$ | $1$ | $1^{6}$ |
| 40.144.1-40.m.1.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.1-40.m.2.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.1-40.p.1.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.1-40.p.2.8 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.3-40.g.1.7 | $40$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 40.144.3-40.t.1.6 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-40.bk.1.8 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.144.3-40.bn.1.6 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.144.3-40.ci.1.8 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-40.ci.2.8 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-40.cl.1.8 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-40.cl.2.8 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 120.144.1-60.k.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-60.k.2.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-60.l.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-60.l.2.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.3-60.eq.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.er.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.fc.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.fd.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.hs.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.hs.2.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.ht.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.ht.2.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.216.7-60.b.1.27 | $120$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 120.288.7-60.gw.1.33 | $120$ | $4$ | $4$ | $7$ | $?$ | not computed |
| 200.360.7-100.b.1.9 | $200$ | $5$ | $5$ | $7$ | $?$ | not computed |
| 120.144.1-120.bk.1.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.bk.2.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.bn.1.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.bn.2.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.3-120.bdw.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bdz.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bgm.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bgp.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cge.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cge.2.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cgh.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cgh.2.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.1-140.d.1.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.1-140.d.2.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.1-140.e.1.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.1-140.e.2.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.3-140.m.1.16 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.n.1.7 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.q.1.16 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.r.1.6 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.u.1.16 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.u.2.14 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.v.1.16 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.v.2.14 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.1-280.m.1.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.1-280.m.2.6 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.1-280.p.1.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.1-280.p.2.6 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.3-280.bk.1.11 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.bn.1.10 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.bw.1.12 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.bz.1.10 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.cu.1.10 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.cu.2.10 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.cx.1.10 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.cx.2.10 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |