Invariants
Level: | $40$ | $\SL_2$-level: | $10$ | Newform level: | $1600$ | ||
Index: | $40$ | $\PSL_2$-index: | $40$ | ||||
Genus: | $1 = 1 + \frac{ 40 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $10^{4}$ | Cusp orbits | $4$ | ||
Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 10H1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.40.1.1 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&32\\39&29\end{bmatrix}$, $\begin{bmatrix}19&12\\3&11\end{bmatrix}$, $\begin{bmatrix}20&27\\11&30\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 40-isogeny field degree: | $72$ |
Cyclic 40-torsion field degree: | $1152$ |
Full 40-torsion field degree: | $18432$ |
Jacobian
Conductor: | $2^{6}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 1600.2.a.w |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + y w + z^{2} + z w + w^{2} $ |
$=$ | $4 x^{2} - 2 y^{2} - 2 y z - y w + z^{2} - z w - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 2 x^{3} z - 8 x^{2} y^{2} + 4 x^{2} z^{2} + 2 x y^{2} z + 3 x z^{3} + 36 y^{4} - 2 y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}x$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 40 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^4\cdot5^2\,\frac{(z+w)^{3}(1656yz^{6}+4380yz^{5}w+3748yz^{4}w^{2}+1952yz^{3}w^{3}+666yz^{2}w^{4}+729yzw^{5}+675yw^{6}-972z^{7}-7920z^{6}w-23226z^{5}w^{2}-39134z^{4}w^{3}-40497z^{3}w^{4}-26118z^{2}w^{5}-9720zw^{6}-1215w^{7})}{80yz^{8}w+160yz^{7}w^{2}-160yz^{6}w^{3}-480yz^{5}w^{4}-300yz^{4}w^{5}+20yz^{3}w^{6}+140yz^{2}w^{7}+60yzw^{8}+5yw^{9}+16z^{10}+80z^{9}w-400z^{7}w^{3}-780z^{6}w^{4}-532z^{5}w^{5}+170z^{4}w^{6}+450z^{3}w^{7}+275z^{2}w^{8}+55zw^{9}-9w^{10}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
5.20.0.b.1 | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
8.2.0.a.1 | $8$ | $20$ | $20$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
5.20.0.b.1 | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.10.0.a.1 | $40$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
40.20.0.f.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.20.1.a.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
40.120.5.g.1 | $40$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
40.120.5.cj.1 | $40$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
40.160.9.n.1 | $40$ | $4$ | $4$ | $9$ | $6$ | $1^{8}$ |
120.120.9.fp.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
120.160.9.z.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
200.200.9.f.1 | $200$ | $5$ | $5$ | $9$ | $?$ | not computed |
280.320.21.n.1 | $280$ | $8$ | $8$ | $21$ | $?$ | not computed |