Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $800$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.1.14 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&6\\20&29\end{bmatrix}$, $\begin{bmatrix}3&34\\4&5\end{bmatrix}$, $\begin{bmatrix}35&6\\33&33\end{bmatrix}$, $\begin{bmatrix}37&30\\11&23\end{bmatrix}$ |
| 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: | $2^{5}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 800.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 275x + 1750 $ |
Rational points
This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^4}{5^4}\cdot\frac{60x^{2}y^{6}-1741875x^{2}y^{4}z^{2}+11070000000x^{2}y^{2}z^{4}-20064384765625x^{2}z^{6}-2450xy^{6}z+44700000xy^{4}z^{3}-236250390625xy^{2}z^{5}+384074902343750xz^{7}-y^{8}+66000y^{6}z^{2}-624062500y^{4}z^{4}+1943992187500y^{2}z^{6}-1834310791015625z^{8}}{z^{2}(x^{2}y^{4}-59000x^{2}y^{2}z^{2}+297250000x^{2}z^{4}-60xy^{4}z+1490000xy^{2}z^{3}-5690000000xz^{5}+2100y^{4}z^{2}-19200000y^{2}z^{4}+27175000000z^{6})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(4)$ | $4$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.12.0.br.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.12.1.h.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 |
|---|---|---|---|---|---|---|
| 40.48.1.o.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.co.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.dk.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.dt.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.ib.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.il.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.ix.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.jl.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.120.9.dv.1 | $40$ | $5$ | $5$ | $9$ | $2$ | $1^{6}\cdot2$ |
| 40.144.9.hz.1 | $40$ | $6$ | $6$ | $9$ | $2$ | $1^{6}\cdot2$ |
| 40.240.17.tl.1 | $40$ | $10$ | $10$ | $17$ | $3$ | $1^{12}\cdot2^{2}$ |
| 120.48.1.bcn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bcv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bej.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.ber.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cgv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.chb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.chz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cij.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.bpb.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.5.pz.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.ben.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ber.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bfd.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bfh.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.boj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bon.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.boz.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bpd.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.13.il.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |