Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
| 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: | $2$ | ||||||
| $\Q$-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: | 40.120.4.90 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&24\\20&13\end{bmatrix}$, $\begin{bmatrix}3&12\\20&29\end{bmatrix}$, $\begin{bmatrix}13&26\\28&13\end{bmatrix}$, $\begin{bmatrix}19&6\\34&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.60.4.d.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{14}\cdot5^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 50.2.a.b$^{2}$, 1600.2.a.b, 1600.2.a.p |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 10 x^{2} + 10 x y + 20 y^{2} - z w + w^{2} $ |
| $=$ | $10 x^{3} - 10 x^{2} y - x z^{2} + x z w + x w^{2} + y z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 250 x^{6} - 125 x^{4} y^{2} + 25 x^{4} y z + 150 x^{4} z^{2} + 20 x^{2} y^{4} - 30 x^{2} y^{3} z + \cdots + 4 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{22510xyz^{8}-43230xyz^{7}w-139400xyz^{6}w^{2}+376940xyz^{5}w^{3}+143650xyz^{4}w^{4}-1196780xyz^{3}w^{5}+1139320xyz^{2}w^{6}-325520xyzw^{7}+33940y^{2}z^{8}-120140y^{2}z^{7}w-106360y^{2}z^{6}w^{2}+1237160y^{2}z^{5}w^{3}-2072800y^{2}z^{4}w^{4}+434490y^{2}z^{3}w^{5}+1744460y^{2}z^{2}w^{6}-1464840y^{2}zw^{7}+325520y^{2}w^{8}+1024z^{10}-6817z^{9}w+18947z^{8}w^{2}-18464z^{7}w^{3}-42554z^{6}w^{4}+150188z^{5}w^{5}-147652z^{4}w^{6}-14656z^{3}w^{7}+126544z^{2}w^{8}-81920zw^{9}+16384w^{10}}{70xyz^{8}-250xyz^{7}w+410xyz^{6}w^{2}-530xyz^{5}w^{3}+350xyz^{4}w^{4}-70xyz^{3}w^{5}-70xyz^{2}w^{6}+20xyzw^{7}+100y^{2}z^{8}-500y^{2}z^{7}w+740y^{2}z^{6}w^{2}-580y^{2}z^{5}w^{3}+300y^{2}z^{4}w^{4}+60y^{2}z^{3}w^{5}-160y^{2}z^{2}w^{6}+90y^{2}zw^{7}-20y^{2}w^{8}-5z^{9}w+29z^{8}w^{2}-59z^{7}w^{3}+53z^{6}w^{4}-17z^{5}w^{5}-13z^{4}w^{6}+16z^{3}w^{7}-4z^{2}w^{8}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.60.4.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 250X^{6}-125X^{4}Y^{2}+25X^{4}YZ+150X^{4}Z^{2}+20X^{2}Y^{4}-30X^{2}Y^{3}Z+10X^{2}YZ^{3}+20X^{2}Z^{4}-4Y^{3}Z^{3}+4Y^{2}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.60.2-10.a.1.1 | $20$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 40.24.0-40.b.1.1 | $40$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.60.2-10.a.1.2 | $40$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.240.8-40.f.1.1 | $40$ | $2$ | $2$ | $8$ | $5$ | $1^{4}$ |
| 40.240.8-40.f.1.2 | $40$ | $2$ | $2$ | $8$ | $5$ | $1^{4}$ |
| 40.240.8-40.g.1.1 | $40$ | $2$ | $2$ | $8$ | $4$ | $1^{4}$ |
| 40.240.8-40.g.1.3 | $40$ | $2$ | $2$ | $8$ | $4$ | $1^{4}$ |
| 40.240.8-40.h.1.2 | $40$ | $2$ | $2$ | $8$ | $2$ | $1^{4}$ |
| 40.240.8-40.h.1.10 | $40$ | $2$ | $2$ | $8$ | $2$ | $1^{4}$ |
| 40.240.8-40.i.1.2 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.240.8-40.i.1.5 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.360.10-40.d.1.8 | $40$ | $3$ | $3$ | $10$ | $3$ | $1^{6}$ |
| 40.480.13-40.bn.1.5 | $40$ | $4$ | $4$ | $13$ | $7$ | $1^{9}$ |
| 120.240.8-120.r.1.14 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.r.1.15 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.t.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.t.1.9 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.x.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.x.1.13 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.z.1.2 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.z.1.9 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.360.14-120.f.1.26 | $120$ | $3$ | $3$ | $14$ | $?$ | not computed |
| 120.480.17-120.fl.1.16 | $120$ | $4$ | $4$ | $17$ | $?$ | not computed |
| 280.240.8-280.v.1.8 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.v.1.14 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.w.1.1 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.w.1.3 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.y.1.2 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.y.1.9 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.z.1.1 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.z.1.13 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |