Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
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: | 8B1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.1.91 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&22\\22&33\end{bmatrix}$, $\begin{bmatrix}13&6\\15&19\end{bmatrix}$, $\begin{bmatrix}21&16\\12&33\end{bmatrix}$, $\begin{bmatrix}37&34\\25&11\end{bmatrix}$ |
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: | $30720$ |
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 $ | $=$ | $ y^{2} - z^{2} - z w + w^{2} $ |
$=$ | $10 x^{2} + 2 y z + y w$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 5 y^{2} z^{2} - 25 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}y$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(3z^{2}+3zw+2w^{2})^{3}}{(2z+w)^{2}(z^{2}+zw-w^{2})^{2}}$ |
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 |
---|---|---|---|---|---|---|
8.12.1.d.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
20.12.0.k.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.12.0.ca.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 |
---|---|---|---|---|---|---|
40.48.1.fc.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.fd.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.fe.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.ff.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.gk.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.gl.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.gm.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.gn.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.120.9.bq.1 | $40$ | $5$ | $5$ | $9$ | $3$ | $1^{6}\cdot2$ |
40.144.9.co.1 | $40$ | $6$ | $6$ | $9$ | $1$ | $1^{6}\cdot2$ |
40.240.17.lw.1 | $40$ | $10$ | $10$ | $17$ | $6$ | $1^{12}\cdot2^{2}$ |
120.48.1.ru.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.rv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.rw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.rx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.ta.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.tb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.tc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.td.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.72.5.cw.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
120.96.5.ca.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
280.48.1.ok.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.ol.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.om.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.on.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.pa.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.pb.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.pc.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.pd.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.192.13.ca.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |