Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $1600$ | ||
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: | 8C1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.1.59 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&36\\4&15\end{bmatrix}$, $\begin{bmatrix}11&22\\15&1\end{bmatrix}$, $\begin{bmatrix}15&2\\1&33\end{bmatrix}$, $\begin{bmatrix}29&6\\36&27\end{bmatrix}$ |
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: | $2^{6}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 1600.2.a.n |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x z + y w $ |
$=$ | $16 x^{2} + 10 y^{2} + 5 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 20 x^{4} + 10 x^{2} y^{2} + x^{2} z^{2} + y^{2} z^{2} $ |
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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
$\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{27000y^{6}-35100y^{4}w^{2}+12510y^{2}w^{4}+4375z^{6}+6750z^{4}w^{2}+3600z^{2}w^{4}-27w^{6}}{1000y^{6}-100y^{4}w^{2}-30y^{2}w^{4}+125z^{6}-50z^{4}w^{2}-20z^{2}w^{4}-w^{6}}$ |
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 |
---|---|---|---|---|---|---|
8.12.0.q.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
20.12.0.n.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.12.1.g.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 |
---|---|---|---|---|---|---|
40.48.1.bk.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.cb.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.cv.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.cz.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.hm.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.hq.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.id.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.ih.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.120.9.dg.1 | $40$ | $5$ | $5$ | $9$ | $4$ | $1^{6}\cdot2$ |
40.144.9.hk.1 | $40$ | $6$ | $6$ | $9$ | $2$ | $1^{6}\cdot2$ |
40.240.17.sw.1 | $40$ | $10$ | $10$ | $17$ | $5$ | $1^{12}\cdot2^{2}$ |
120.48.1.bba.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.bbi.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.bbq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.bby.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.cfi.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.cfq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.cfz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.cgh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.72.5.boe.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
120.96.5.pc.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
280.48.1.bcw.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bda.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bdm.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bdq.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bms.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bmw.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bni.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bnm.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.192.13.hw.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |