Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $400$ | ||
| Index: | $120$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $4 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $10^{2}\cdot20^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20A4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.120.4.212 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&8\\22&21\end{bmatrix}$, $\begin{bmatrix}15&16\\6&17\end{bmatrix}$, $\begin{bmatrix}31&36\\26&39\end{bmatrix}$, $\begin{bmatrix}39&26\\30&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.60.4.c.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{10}\cdot5^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 50.2.a.b$^{2}$, 400.2.a.d, 400.2.a.h |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 10 x^{2} + 5 x y + 5 y^{2} - z w + w^{2} $ |
| $=$ | $5 x y^{2} - x z^{2} + x z w - 5 y^{3} - y z^{2} + 3 y z w - y w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 500 x^{6} - 125 x^{4} y^{2} + 25 x^{4} y z + 150 x^{4} z^{2} + 10 x^{2} y^{4} - 15 x^{2} y^{3} z + \cdots + y^{2} z^{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.
| Canonical model |
|---|
| $(0:0:1:0)$, $(0:0:1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{28225xyz^{8}-81685xyz^{7}w-122880xyz^{6}w^{2}+807050xyz^{5}w^{3}-964575xyz^{4}w^{4}-381145xyz^{3}w^{5}+1441890xyz^{2}w^{6}-895180xyzw^{7}+162760xyw^{8}+5715y^{2}z^{8}-38455y^{2}z^{7}w+16520y^{2}z^{6}w^{2}+430110y^{2}z^{5}w^{3}-1108225y^{2}z^{4}w^{4}+815635y^{2}z^{3}w^{5}+302570y^{2}z^{2}w^{6}-569660y^{2}zw^{7}+162760y^{2}w^{8}-2048z^{10}+12491z^{9}w-29060z^{8}w^{2}+25933z^{7}w^{3}+2390z^{6}w^{4}+7291z^{5}w^{5}-89468z^{4}w^{6}+131925z^{3}w^{7}-78642z^{2}w^{8}+17356zw^{9}-216w^{10}}{85xyz^{8}-375xyz^{7}w+575xyz^{6}w^{2}-555xyz^{5}w^{3}+325xyz^{4}w^{4}-5xyz^{3}w^{5}-115xyz^{2}w^{6}+55xyzw^{7}-10xyw^{8}+15y^{2}z^{8}-125y^{2}z^{7}w+165y^{2}z^{6}w^{2}-25y^{2}z^{5}w^{3}-25y^{2}z^{4}w^{4}+65y^{2}z^{3}w^{5}-45y^{2}z^{2}w^{6}+35y^{2}zw^{7}-10y^{2}w^{8}+7z^{9}w-30z^{8}w^{2}+60z^{7}w^{3}-68z^{6}w^{4}+34z^{5}w^{5}+8z^{4}w^{6}-10z^{3}w^{7}-8z^{2}w^{8}+9zw^{9}-2w^{10}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.60.4.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 500X^{6}-125X^{4}Y^{2}+25X^{4}YZ+150X^{4}Z^{2}+10X^{2}Y^{4}-15X^{2}Y^{3}Z+5X^{2}YZ^{3}+10X^{2}Z^{4}-Y^{3}Z^{3}+Y^{2}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.24.0-20.a.1.3 | $40$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.60.2-10.a.1.2 | $40$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 40.60.2-10.a.1.3 | $40$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.240.8-20.c.1.1 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.240.8-20.e.1.7 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-20.e.1.8 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-40.e.1.1 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.240.8-40.e.1.7 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.240.8-40.i.1.3 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.240.8-40.i.1.5 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.360.10-20.c.1.7 | $40$ | $3$ | $3$ | $10$ | $2$ | $1^{6}$ |
| 40.480.13-20.m.1.2 | $40$ | $4$ | $4$ | $13$ | $2$ | $1^{9}$ |
| 120.240.8-60.g.1.4 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-60.g.1.6 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-60.i.1.5 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-60.i.1.7 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.p.1.3 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.p.1.16 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.v.1.4 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.v.1.14 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.360.14-60.e.1.5 | $120$ | $3$ | $3$ | $14$ | $?$ | not computed |
| 120.480.17-60.e.1.16 | $120$ | $4$ | $4$ | $17$ | $?$ | not computed |
| 280.240.8-140.g.1.3 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-140.g.1.8 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-140.h.1.5 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-140.h.1.8 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.o.1.8 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.o.1.13 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.r.1.8 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.r.1.13 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |