Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
Index: | $96$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $4^{2}\cdot8$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8K1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.1.1278 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&1\\8&37\end{bmatrix}$, $\begin{bmatrix}21&29\\18&39\end{bmatrix}$, $\begin{bmatrix}39&23\\38&9\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 40-isogeny field degree: | $12$ |
Cyclic 40-torsion field degree: | $192$ |
Full 40-torsion field degree: | $7680$ |
Jacobian
Conductor: | $2^{5}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 5 x^{2} - 5 y^{2} + z^{2} - z w - w^{2} $ |
$=$ | $4 x^{2} + 12 x y + 4 y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 8 x^{3} z + 200 x^{2} y^{2} + 34 x^{2} z^{2} - 100 x y^{2} z + 72 x z^{3} - 125 y^{4} + 81 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^2}{5^4}\cdot\frac{(z^{2}-6zw-w^{2})^{3}(3z^{2}+2zw-3w^{2})^{3}(13z^{4}-16z^{3}w+2z^{2}w^{2}+16zw^{3}+13w^{4})^{3}}{(z^{2}+w^{2})^{8}(z^{2}-zw-w^{2})^{4}}$ |
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 |
---|---|---|---|---|---|---|
40.48.1.fu.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 |
---|---|---|---|---|---|---|
40.480.33.boq.1 | $40$ | $5$ | $5$ | $33$ | $4$ | $1^{14}\cdot2^{5}\cdot4^{2}$ |
40.576.33.cel.1 | $40$ | $6$ | $6$ | $33$ | $1$ | $1^{14}\cdot2\cdot4^{4}$ |
40.960.65.dfl.2 | $40$ | $10$ | $10$ | $65$ | $8$ | $1^{28}\cdot2^{6}\cdot4^{6}$ |
120.288.17.bfbj.2 | $120$ | $3$ | $3$ | $17$ | $?$ | not computed |
120.384.17.fir.1 | $120$ | $4$ | $4$ | $17$ | $?$ | not computed |