Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
| Index: | $120$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $5 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $5^{8}\cdot20^{4}$ | Cusp orbits | $4\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $4$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-8$) |
Other labels
| Cummins and Pauli (CP) label: | 20E5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.120.5.111 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}13&35\\2&7\end{bmatrix}$, $\begin{bmatrix}13&37\\4&7\end{bmatrix}$, $\begin{bmatrix}17&21\\4&21\end{bmatrix}$, $\begin{bmatrix}33&36\\8&7\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{23}\cdot5^{10}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{5}$ |
| Newforms: | 50.2.a.b, 400.2.a.d, 1600.2.a.c, 1600.2.a.p, 1600.2.a.q |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 5 x y + 10 x z - w t $ |
| $=$ | $10 x^{2} - 2 y^{2} + 2 y z + 2 z^{2} - w^{2}$ | |
| $=$ | $5 y^{2} - 10 y z + 10 z^{2} - 2 w^{2} - 2 t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{8} - 2 x^{6} y^{2} - 30 x^{6} z^{2} + x^{4} y^{4} + 50 x^{4} y^{2} z^{2} + 285 x^{4} z^{4} + \cdots + 900 z^{8} $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{10}w$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 40.60.3.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -2x-y+z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x-y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x+z$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{3}Y+7X^{2}Y^{2}-XY^{3}+Y^{4}+6XY^{2}Z-2Y^{3}Z+2X^{2}Z^{2}-6XYZ^{2}-4Y^{2}Z^{2}-12XZ^{3}+4YZ^{3}+4Z^{4} $ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.60.2.h.1 | $20$ | $2$ | $2$ | $2$ | $1$ | $1^{3}$ |
| 40.60.2.a.1 | $40$ | $2$ | $2$ | $2$ | $1$ | $1^{3}$ |
| 40.60.3.c.1 | $40$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.240.13.c.1 | $40$ | $2$ | $2$ | $13$ | $7$ | $1^{8}$ |
| 40.240.13.eh.1 | $40$ | $2$ | $2$ | $13$ | $6$ | $1^{8}$ |
| 40.240.13.ek.1 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{8}$ |
| 40.240.13.eo.1 | $40$ | $2$ | $2$ | $13$ | $10$ | $1^{8}$ |
| 40.240.13.iu.1 | $40$ | $2$ | $2$ | $13$ | $6$ | $1^{8}$ |
| 40.240.13.iw.1 | $40$ | $2$ | $2$ | $13$ | $7$ | $1^{8}$ |
| 40.240.13.ji.1 | $40$ | $2$ | $2$ | $13$ | $8$ | $1^{8}$ |
| 40.240.13.jk.1 | $40$ | $2$ | $2$ | $13$ | $6$ | $1^{8}$ |
| 40.360.13.bm.1 | $40$ | $3$ | $3$ | $13$ | $6$ | $1^{8}$ |
| 120.240.13.baa.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.240.13.bac.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.240.13.bao.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.240.13.baq.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.240.13.bku.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.240.13.bkw.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.240.13.bli.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.240.13.blk.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.240.13.bej.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.240.13.bek.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.240.13.beq.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.240.13.ber.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.240.13.bir.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.240.13.bis.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.240.13.biy.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.240.13.biz.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |