Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $800$ | ||
Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8F1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.1.127 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&29\\32&5\end{bmatrix}$, $\begin{bmatrix}27&12\\12&11\end{bmatrix}$, $\begin{bmatrix}35&22\\18&25\end{bmatrix}$ |
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: | $15360$ |
Jacobian
Conductor: | $2^{5}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 800.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - y^{2} + y z - z^{2} + 2 w^{2} $ |
$=$ | $2 x^{2} + 3 y^{2} + 2 y z + 3 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 20 x^{2} z^{2} + y^{4} + 60 y^{2} z^{2} + 900 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^4\cdot5^2\,\frac{z^{3}(5z^{2}-4w^{2})(300yz^{4}w^{2}-240yz^{2}w^{4}-64yw^{6}-125z^{7}+200z^{5}w^{2}+160z^{3}w^{4}-192zw^{6})}{w^{8}(20yzw^{2}-25z^{4}+20z^{2}w^{2}+4w^{4})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
8.24.0.u.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.bh.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.dz.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.ea.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.1.p.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
40.24.1.do.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
40.24.1.dr.1 | $40$ | $2$ | $2$ | $1$ | $1$ | 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 |
---|---|---|---|---|---|---|
40.240.17.ie.1 | $40$ | $5$ | $5$ | $17$ | $5$ | $1^{14}\cdot2$ |
40.288.17.wc.1 | $40$ | $6$ | $6$ | $17$ | $9$ | $1^{14}\cdot2$ |
40.480.33.bku.1 | $40$ | $10$ | $10$ | $33$ | $9$ | $1^{28}\cdot2^{2}$ |
120.144.9.eas.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
120.192.9.bki.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |