Invariants
Level: | $40$ | $\SL_2$-level: | $10$ | Newform level: | $20$ | ||
Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $2^{6}\cdot10^{6}$ | Cusp orbits | $2^{6}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 10K1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.72.1.54 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&2\\22&31\end{bmatrix}$, $\begin{bmatrix}15&18\\24&29\end{bmatrix}$, $\begin{bmatrix}15&28\\18&35\end{bmatrix}$, $\begin{bmatrix}27&14\\24&7\end{bmatrix}$, $\begin{bmatrix}31&12\\22&21\end{bmatrix}$, $\begin{bmatrix}33&14\\20&7\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 40.144.1-40.a.2.1, 40.144.1-40.a.2.2, 40.144.1-40.a.2.3, 40.144.1-40.a.2.4, 40.144.1-40.a.2.5, 40.144.1-40.a.2.6, 40.144.1-40.a.2.7, 40.144.1-40.a.2.8, 120.144.1-40.a.2.1, 120.144.1-40.a.2.2, 120.144.1-40.a.2.3, 120.144.1-40.a.2.4, 120.144.1-40.a.2.5, 120.144.1-40.a.2.6, 120.144.1-40.a.2.7, 120.144.1-40.a.2.8, 280.144.1-40.a.2.1, 280.144.1-40.a.2.2, 280.144.1-40.a.2.3, 280.144.1-40.a.2.4, 280.144.1-40.a.2.5, 280.144.1-40.a.2.6, 280.144.1-40.a.2.7, 280.144.1-40.a.2.8 |
Cyclic 40-isogeny field degree: | $4$ |
Cyclic 40-torsion field degree: | $64$ |
Full 40-torsion field degree: | $10240$ |
Jacobian
Conductor: | $2^{2}\cdot5$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 20.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 10 x^{2} - 3 z^{2} - 2 z w + w^{2} $ |
$=$ | $10 x y - 10 y^{2} - 4 z^{2} + z w$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 50 x^{4} + 20 x^{2} y^{2} - 40 x^{2} y z + 25 x^{2} z^{2} + 2 y^{2} z^{2} - 8 y z^{3} + 8 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
$\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{(179z^{6}-146z^{5}w-135z^{4}w^{2}+180z^{3}w^{3}-75z^{2}w^{4}+14zw^{5}-w^{6})^{3}}{z^{10}(z+w)(3z-w)^{5}(4z-w)^{2}}$ |
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 |
---|---|---|---|---|---|---|
10.36.1.a.1 | $10$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.24.1.bw.2 | $40$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
40.36.0.a.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.36.0.f.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
40.144.5.b.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.c.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.e.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.f.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.n.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.o.2 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.q.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.r.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.360.13.a.1 | $40$ | $5$ | $5$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
120.144.5.cv.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cw.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cy.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cz.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ef.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eg.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ei.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ej.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.216.13.a.1 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.288.13.oq.1 | $120$ | $4$ | $4$ | $13$ | $?$ | not computed |
200.360.13.a.2 | $200$ | $5$ | $5$ | $13$ | $?$ | not computed |
280.144.5.b.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.c.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.e.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.f.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.n.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.o.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.q.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.r.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |