Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $360$ | $\PSL_2$-index: | $180$ | ||||
| Genus: | $10 = 1 + \frac{ 180 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $5^{6}\cdot10^{3}\cdot40^{3}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $5$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 7$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 7$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4,-16$) |
Other labels
| Cummins and Pauli (CP) label: | 40D10 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.360.10.855 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&6\\20&1\end{bmatrix}$, $\begin{bmatrix}3&17\\4&27\end{bmatrix}$, $\begin{bmatrix}3&21\\4&37\end{bmatrix}$, $\begin{bmatrix}3&36\\4&7\end{bmatrix}$, $\begin{bmatrix}17&33\\0&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.180.10.cx.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $32$ |
| Full 40-torsion field degree: | $2048$ |
Jacobian
| Conductor: | $2^{42}\cdot5^{17}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{10}$ |
| Newforms: | 20.2.a.a, 50.2.a.b$^{2}$, 100.2.a.a, 320.2.a.c, 320.2.a.f, 1600.2.a.a, 1600.2.a.c, 1600.2.a.o, 1600.2.a.q |
Models
Canonical model in $\mathbb{P}^{ 9 }$ defined by 28 equations
| $ 0 $ | $=$ | $ 2 x z - 2 x w + r s + r a $ |
| $=$ | $2 y^{2} + 2 y z - v r$ | |
| $=$ | $2 y z + 2 y w + 2 z^{2} + 2 z w - v s + v a$ | |
| $=$ | $2 y z + 2 z^{2} + t u - t a - u s - u a + v a - s^{2} + a^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{10} y^{2} + 3 x^{10} y z + x^{10} z^{2} - 2 x^{8} y^{4} + 3 x^{8} y^{3} z + 17 x^{8} y^{2} z^{2} + \cdots - 16 z^{12} $ |
Rational points
This modular curve has 2 rational cusps and 2 rational CM points, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:0:0:0:0:1:1)$, $(0:0:0:0:1:2:0:0:0:1)$ |
Maps to other modular curves
Map of degree 3 from the canonical model of this modular curve to the canonical model of the modular curve 40.60.4.br.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle s$ |
| $\displaystyle W$ | $=$ | $\displaystyle t+v$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{2}-2XY+4Y^{2}-ZW $ |
| $=$ | $ 2X^{3}+4X^{2}Y+2XY^{2}+2XZ^{2}+2YZ^{2}-2XZW-YW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.180.10.cx.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}a$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}r$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{10}Y^{2}-2X^{8}Y^{4}+X^{6}Y^{6}+3X^{10}YZ+3X^{8}Y^{3}Z-8X^{6}Y^{5}Z+2X^{4}Y^{7}Z+X^{10}Z^{2}+17X^{8}Y^{2}Z^{2}-X^{6}Y^{4}Z^{2}-6X^{4}Y^{6}Z^{2}+19X^{8}YZ^{3}-11X^{6}Y^{3}Z^{3}+10X^{4}Y^{5}Z^{3}-8X^{2}Y^{7}Z^{3}+8X^{8}Z^{4}+40X^{6}Y^{2}Z^{4}-22X^{4}Y^{4}Z^{4}+12X^{2}Y^{6}Z^{4}+77X^{6}YZ^{5}-150X^{4}Y^{3}Z^{5}+112X^{2}Y^{5}Z^{5}+26X^{6}Z^{6}+58X^{4}Y^{2}Z^{6}-68X^{2}Y^{4}Z^{6}+16Y^{6}Z^{6}+310X^{4}YZ^{7}-436X^{2}Y^{3}Z^{7}+16Y^{5}Z^{7}+134X^{4}Z^{8}-104X^{2}Y^{2}Z^{8}-64Y^{4}Z^{8}+564X^{2}YZ^{9}-48Y^{3}Z^{9}+376X^{2}Z^{10}+64Y^{2}Z^{10}+16YZ^{11}-16Z^{12} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(5)$ | $5$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 8.24.0-8.p.1.7 | $8$ | $15$ | $15$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.120.4-40.br.1.13 | $40$ | $3$ | $3$ | $4$ | $2$ | $1^{6}$ |
| 40.180.4-20.c.1.4 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{6}$ |
| 40.180.4-20.c.1.22 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.720.19-40.ov.1.19 | $40$ | $2$ | $2$ | $19$ | $6$ | $1^{9}$ |
| 40.720.19-40.ox.1.5 | $40$ | $2$ | $2$ | $19$ | $9$ | $1^{9}$ |
| 40.720.19-40.pd.1.11 | $40$ | $2$ | $2$ | $19$ | $12$ | $1^{9}$ |
| 40.720.19-40.pf.1.7 | $40$ | $2$ | $2$ | $19$ | $6$ | $1^{9}$ |
| 40.720.19-40.qf.1.17 | $40$ | $2$ | $2$ | $19$ | $9$ | $1^{9}$ |
| 40.720.19-40.qh.1.8 | $40$ | $2$ | $2$ | $19$ | $7$ | $1^{9}$ |
| 40.720.19-40.qn.1.9 | $40$ | $2$ | $2$ | $19$ | $7$ | $1^{9}$ |
| 40.720.19-40.qp.1.6 | $40$ | $2$ | $2$ | $19$ | $9$ | $1^{9}$ |
| 40.720.22-40.cg.1.16 | $40$ | $2$ | $2$ | $22$ | $6$ | $1^{12}$ |
| 40.720.22-40.cp.1.8 | $40$ | $2$ | $2$ | $22$ | $6$ | $1^{12}$ |
| 40.720.22-40.fd.1.10 | $40$ | $2$ | $2$ | $22$ | $12$ | $1^{12}$ |
| 40.720.22-40.fe.1.16 | $40$ | $2$ | $2$ | $22$ | $12$ | $1^{12}$ |
| 40.720.22-40.gb.1.3 | $40$ | $2$ | $2$ | $22$ | $8$ | $1^{12}$ |
| 40.720.22-40.gd.1.3 | $40$ | $2$ | $2$ | $22$ | $8$ | $1^{12}$ |
| 40.720.22-40.gn.1.11 | $40$ | $2$ | $2$ | $22$ | $10$ | $1^{12}$ |
| 40.720.22-40.gp.1.7 | $40$ | $2$ | $2$ | $22$ | $10$ | $1^{12}$ |
| 40.720.22-40.ht.1.11 | $40$ | $2$ | $2$ | $22$ | $11$ | $1^{12}$ |
| 40.720.22-40.hv.1.13 | $40$ | $2$ | $2$ | $22$ | $11$ | $1^{12}$ |
| 40.720.22-40.ib.1.3 | $40$ | $2$ | $2$ | $22$ | $8$ | $1^{12}$ |
| 40.720.22-40.id.1.9 | $40$ | $2$ | $2$ | $22$ | $8$ | $1^{12}$ |
| 40.720.22-40.iz.1.4 | $40$ | $2$ | $2$ | $22$ | $7$ | $1^{12}$ |
| 40.720.22-40.jb.1.9 | $40$ | $2$ | $2$ | $22$ | $7$ | $1^{12}$ |
| 40.720.22-40.jh.1.12 | $40$ | $2$ | $2$ | $22$ | $11$ | $1^{12}$ |
| 40.720.22-40.jj.1.13 | $40$ | $2$ | $2$ | $22$ | $11$ | $1^{12}$ |