Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $20$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (all of which are rational) | Cusp widths | $2^{3}\cdot10^{3}$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.72.1.49 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&16\\34&25\end{bmatrix}$, $\begin{bmatrix}7&16\\16&7\end{bmatrix}$, $\begin{bmatrix}9&18\\18&29\end{bmatrix}$, $\begin{bmatrix}9&34\\0&23\end{bmatrix}$, $\begin{bmatrix}33&26\\0&39\end{bmatrix}$, $\begin{bmatrix}33&36\\10&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 10.36.1.a.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: | $2^{2}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 20.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} + 4x + 4 $ |
Rational points
This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(-1:0:1)$, $(0:-2:1)$, $(4:-10:1)$, $(4:10:1)$, $(0:2:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{702x^{2}y^{10}+172801470x^{2}y^{8}z^{2}-7293296040x^{2}y^{6}z^{4}+58873460634x^{2}y^{4}z^{6}-140838145082x^{2}y^{2}z^{8}+63165448681x^{2}z^{10}+167859xy^{10}z+671160132xy^{8}z^{3}-9162527235xy^{6}z^{5}-2073715896xy^{4}z^{7}+238834250645xy^{2}z^{9}-483993059328xz^{11}+y^{12}+14328537y^{10}z^{2}-537229632y^{8}z^{4}+16163936765y^{6}z^{6}-140965016207y^{4}z^{8}+465116093315y^{2}z^{10}-516640929884z^{12}}{x^{2}y^{10}-6880x^{2}y^{8}z^{2}-27648x^{2}y^{6}z^{4}-3958272x^{2}y^{4}z^{6}-53063680x^{2}y^{2}z^{8}+33021952x^{2}z^{10}-40xy^{10}z+32720xy^{8}z^{3}+1071360xy^{6}z^{5}+11865344xy^{4}z^{7}-72558592xy^{2}z^{9}-280088576xz^{11}+700y^{10}z^{2}-101968y^{8}z^{4}-1150976y^{6}z^{6}+13327104y^{4}z^{8}+49731584y^{2}z^{10}-313110528z^{12}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.12.0-2.a.1.1 | $40$ | $6$ | $6$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.144.1-10.a.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.1-10.a.2.5 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.360.7-10.a.1.1 | $40$ | $5$ | $5$ | $7$ | $0$ | $1^{6}$ |
| 40.144.1-20.a.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.1-20.a.2.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.3-20.a.1.3 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-20.a.1.14 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-20.b.1.3 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-20.b.1.32 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-20.b.1.35 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-20.c.1.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.144.3-20.c.1.3 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.144.3-20.c.1.6 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.144.3-20.d.1.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-20.d.1.4 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-20.d.1.6 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-20.e.1.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-20.e.1.3 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-20.e.1.7 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-20.e.2.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-20.e.2.4 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-20.e.2.7 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-20.f.1.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-20.f.1.4 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-20.f.1.6 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-20.f.2.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-20.f.2.3 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-20.f.2.7 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 120.144.1-30.b.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-30.b.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.216.7-30.a.1.15 | $120$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 120.288.7-30.a.1.7 | $120$ | $4$ | $4$ | $7$ | $?$ | not computed |
| 40.144.1-40.a.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.1-40.a.2.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.1-40.b.1.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.1-40.b.2.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.144.3-40.a.1.6 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.144.3-40.a.1.8 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.144.3-40.a.1.15 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.144.3-40.b.1.4 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.144.3-40.b.1.6 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.144.3-40.b.1.9 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.144.3-40.c.1.2 | $40$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 40.144.3-40.c.1.8 | $40$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 40.144.3-40.c.1.15 | $40$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 40.144.3-40.d.1.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-40.d.1.6 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-40.d.1.9 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.144.3-40.e.1.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-40.e.1.8 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-40.e.1.12 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-40.e.2.4 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-40.e.2.7 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-40.e.2.12 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-40.f.1.4 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-40.f.1.5 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-40.f.1.9 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-40.f.2.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-40.f.2.7 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.144.3-40.f.2.9 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 200.360.7-50.a.1.2 | $200$ | $5$ | $5$ | $7$ | $?$ | not computed |
| 120.144.1-60.b.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-60.b.2.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.3-60.a.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.a.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.a.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.b.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.b.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.b.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.c.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.c.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.c.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.d.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.d.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.d.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.ca.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.ca.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.ca.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.ca.2.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.ca.2.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.ca.2.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.cb.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.cb.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.cb.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.cb.2.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.cb.2.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.cb.2.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.1-70.a.1.4 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.1-70.a.2.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.c.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.c.2.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.d.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.1-120.d.2.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.3-120.a.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.a.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.a.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.b.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.b.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.b.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.c.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.c.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.c.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.d.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.d.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.d.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.fs.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.fs.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.fs.1.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.fs.2.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.fs.2.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.fs.2.26 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.ft.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.ft.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.ft.1.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.ft.2.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.ft.2.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.ft.2.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.1-140.a.1.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.1-140.a.2.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.3-140.a.1.3 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.a.1.7 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.a.1.14 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.b.1.3 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.b.1.5 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.b.1.10 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.c.1.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.c.1.7 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.c.1.14 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.d.1.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.d.1.5 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.d.1.10 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.e.1.3 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.e.1.7 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.e.1.11 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.e.2.4 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.e.2.7 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.e.2.11 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.f.1.4 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.f.1.5 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.f.1.9 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.f.2.3 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.f.2.6 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-140.f.2.9 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.1-280.a.1.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.1-280.a.2.6 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.1-280.b.1.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.1-280.b.2.8 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.144.3-280.a.1.5 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.a.1.15 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.a.1.28 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.b.1.3 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.b.1.9 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.b.1.18 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.c.1.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.c.1.15 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.c.1.28 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.d.1.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.d.1.9 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.d.1.18 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.e.1.10 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.e.1.11 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.e.1.26 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.e.2.12 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.e.2.13 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.e.2.26 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.f.1.6 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.f.1.9 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.f.1.17 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.f.2.7 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.f.2.10 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.144.3-280.f.2.17 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |