Invariants
Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $2^{4}\cdot4^{4}\cdot10^{4}\cdot20^{4}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2 \le \gamma \le 4$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 20I5 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.288.5.174 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&34\\28&13\end{bmatrix}$, $\begin{bmatrix}11&3\\0&29\end{bmatrix}$, $\begin{bmatrix}13&11\\36&33\end{bmatrix}$, $\begin{bmatrix}17&24\\36&25\end{bmatrix}$, $\begin{bmatrix}23&11\\12&7\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.144.5.fm.1 for the level structure with $-I$) |
Cyclic 40-isogeny field degree: | $2$ |
Cyclic 40-torsion field degree: | $16$ |
Full 40-torsion field degree: | $2560$ |
Jacobian
Conductor: | $2^{26}\cdot5^{7}$ |
Simple: | no |
Squarefree: | yes |
Decomposition: | $1^{3}\cdot2$ |
Newforms: | 20.2.a.a, 320.2.c.b, 1600.2.a.k, 1600.2.a.w |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
$ 0 $ | $=$ | $ y z - z^{2} + w^{2} $ |
$=$ | $y w - w^{2} + t^{2}$ | |
$=$ | $10 x^{2} + y^{2} + 3 y w + w^{2} - t^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 8 x^{6} z + 40 x^{5} y^{2} + 24 x^{5} z^{2} + 100 x^{4} y^{2} z + 18 x^{4} z^{3} + 40 x^{3} y^{2} z^{2} + \cdots - 5 y^{2} z^{5} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Canonical model |
---|
$(0:0:-1:1:1)$, $(0:0:-1:-1:1)$, $(0:0:1:1:1)$, $(0:0:1:-1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{y^{18}-12y^{16}t^{2}+60y^{14}t^{4}-184y^{12}t^{6}+492y^{10}t^{8}-1368y^{8}t^{10}+3736y^{6}t^{12}-10320y^{4}t^{14}+30156y^{2}t^{16}+124w^{18}-720w^{16}t^{2}+5220w^{14}t^{4}-18720w^{12}t^{6}+64560w^{10}t^{8}-148320w^{8}t^{10}+281360w^{6}t^{12}-381600w^{4}t^{14}+294600w^{2}t^{16}-92408t^{18}}{t^{4}w^{2}(w-t)^{5}(w+t)^{5}(5w^{2}-t^{2})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.144.5.fm.1 :
$\displaystyle X$ | $=$ | $\displaystyle w-t$ |
$\displaystyle Y$ | $=$ | $\displaystyle x$ |
$\displaystyle Z$ | $=$ | $\displaystyle z+t$ |
Equation of the image curve:
$0$ | $=$ | $ 40X^{5}Y^{2}+8X^{6}Z+100X^{4}Y^{2}Z-50X^{2}Y^{4}Z+24X^{5}Z^{2}+40X^{3}Y^{2}Z^{2}-100XY^{4}Z^{2}+18X^{4}Z^{3}-60X^{2}Y^{2}Z^{3}-50Y^{4}Z^{3}-4X^{3}Z^{4}-40XY^{2}Z^{4}-8X^{2}Z^{5}-5Y^{2}Z^{5}-2XZ^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
20.144.1-20.f.2.11 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
40.144.1-20.f.2.27 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
40.144.1-40.ba.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
40.144.1-40.ba.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
40.144.3-40.ee.2.9 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
40.144.3-40.ee.2.14 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
40.576.9-40.bb.1.7 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
40.576.9-40.bb.3.6 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
40.576.9-40.bc.2.7 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
40.576.9-40.bc.4.7 | $40$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
40.576.13-40.nk.2.15 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
40.576.13-40.nl.2.15 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
40.576.13-40.ny.2.15 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{4}\cdot2^{2}$ |
40.576.13-40.oa.1.22 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{4}\cdot2^{2}$ |
40.576.13-40.pq.1.14 | $40$ | $2$ | $2$ | $13$ | $0$ | $1^{4}\cdot2^{2}$ |
40.576.13-40.ps.2.15 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{4}\cdot2^{2}$ |
40.576.13-40.qy.1.13 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
40.576.13-40.qz.1.13 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
40.1440.37-40.lf.1.2 | $40$ | $5$ | $5$ | $37$ | $5$ | $1^{16}\cdot2^{8}$ |