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: | $4$ (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: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.1.57 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&22\\4&5\end{bmatrix}$, $\begin{bmatrix}15&38\\1&1\end{bmatrix}$, $\begin{bmatrix}19&18\\20&9\end{bmatrix}$, $\begin{bmatrix}35&16\\18&3\end{bmatrix}$ |
| 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 1 stored non-cuspidal point.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{3780x^{2}y^{6}+2206875x^{2}y^{4}z^{2}+783444375000x^{2}y^{2}z^{4}+3043839736328125x^{2}z^{6}-142650xy^{6}z+825600000xy^{4}z^{3}+18687145703125xy^{2}z^{5}+58265602636718750xz^{7}-27y^{8}-262000y^{6}z^{2}+30340000000y^{4}z^{4}+212977445312500y^{2}z^{6}+278272046142578125z^{8}}{100x^{2}y^{6}+2629375x^{2}y^{4}z^{2}+625000x^{2}y^{2}z^{4}+9765625x^{2}z^{6}+4750xy^{6}z+50550000xy^{4}z^{3}-5859375xy^{2}z^{5}-97656250xz^{7}+y^{8}+126000y^{6}z^{2}+243000000y^{4}z^{4}-117187500y^{2}z^{6}-1708984375z^{8}}$ |
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.v.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.12.0.bt.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.bo.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.cq.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.ep.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.es.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.iz.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.jl.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.jo.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.48.1.ka.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.120.9.el.1 | $40$ | $5$ | $5$ | $9$ | $6$ | $1^{6}\cdot2$ |
| 40.144.9.jf.1 | $40$ | $6$ | $6$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 40.240.17.vh.1 | $40$ | $10$ | $10$ | $17$ | $10$ | $1^{12}\cdot2^{2}$ |
| 80.48.2.db.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.dd.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.dz.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.eb.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.ex.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.ez.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.ff.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.fh.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.48.1.bdt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.beb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bfp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bfx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cir.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.ciz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cjw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cke.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.btb.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.5.rn.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 240.48.2.gd.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.gf.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.hb.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.hd.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.hz.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.ib.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.ih.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.ij.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 280.48.1.bgj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bgn.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bhp.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bht.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bqf.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bqj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.brl.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.brp.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.13.kh.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |