Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $1600$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $8^{24}$ | Cusp orbits | $4^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8A5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.384.5.319 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&2\\4&17\end{bmatrix}$, $\begin{bmatrix}3&10\\16&13\end{bmatrix}$, $\begin{bmatrix}19&0\\0&39\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.192.5.c.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $1920$ |
Jacobian
| Conductor: | $2^{28}\cdot5^{4}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 32.2.a.a, 64.2.b.a, 800.2.a.d, 1600.2.a.n |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} + y^{2} + z^{2} $ |
| $=$ | $5 x y - w^{2}$ | |
| $=$ | $5 x^{2} - 5 y^{2} + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 625 x^{8} + 250 x^{6} z^{2} + 350 x^{4} y^{4} + 125 x^{4} z^{4} + 150 x^{2} y^{4} z^{2} + 40 x^{2} z^{6} + \cdots + 4 z^{8} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
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 40.96.3.w.2 :
| $\displaystyle X$ | $=$ | $\displaystyle 2w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -z-t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-6Y^{4}+2Y^{3}Z+6Y^{2}Z^{2}+8YZ^{3}+4Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.192.5.c.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x+z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 625X^{8}+250X^{6}Z^{2}+350X^{4}Y^{4}+125X^{4}Z^{4}+150X^{2}Y^{4}Z^{2}+40X^{2}Z^{6}+81Y^{8}+36Y^{4}Z^{4}+4Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.192.3-8.d.1.4 | $8$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.192.1-40.a.1.7 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 40.192.1-40.a.1.10 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 40.192.1-40.b.2.2 | $40$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 40.192.1-40.b.2.9 | $40$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 40.192.1-40.m.2.1 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 40.192.1-40.m.2.6 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 40.192.3-8.d.1.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.192.3-40.r.2.12 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.192.3-40.r.2.13 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.192.3-40.s.2.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.192.3-40.s.2.10 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.192.3-40.w.2.6 | $40$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 40.192.3-40.w.2.13 | $40$ | $2$ | $2$ | $3$ | $1$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.1920.69-40.bb.2.2 | $40$ | $5$ | $5$ | $69$ | $14$ | $1^{26}\cdot2^{15}\cdot4^{2}$ |
| 40.2304.73-40.lc.2.4 | $40$ | $6$ | $6$ | $73$ | $9$ | $1^{28}\cdot2^{4}\cdot4^{8}$ |
| 40.3840.137-40.ho.1.7 | $40$ | $10$ | $10$ | $137$ | $21$ | $1^{54}\cdot2^{19}\cdot4^{10}$ |