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: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.1.56 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&10\\8&37\end{bmatrix}$, $\begin{bmatrix}9&4\\24&5\end{bmatrix}$, $\begin{bmatrix}15&16\\17&17\end{bmatrix}$, $\begin{bmatrix}25&16\\21&23\end{bmatrix}$ |
| 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: | $2^{6}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1600.2.a.n |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 1100x + 14000 $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{7560x^{2}y^{6}-35310000x^{2}y^{4}z^{2}+100280880000000x^{2}y^{2}z^{4}-3116891890000000000x^{2}z^{6}+570600xy^{6}z+26419200000xy^{4}z^{3}-4783909300000000xy^{2}z^{5}+119327954200000000000xz^{7}+27y^{8}-2096000y^{6}z^{2}-1941760000000y^{4}z^{4}+109044452000000000y^{2}z^{6}-1139802301000000000000z^{8}}{200x^{2}y^{6}-42070000x^{2}y^{4}z^{2}+80000000x^{2}y^{2}z^{4}-10000000000x^{2}z^{6}-19000xy^{6}z+1617600000xy^{4}z^{3}+1500000000xy^{2}z^{5}-200000000000xz^{7}-y^{8}+1008000y^{6}z^{2}-15552000000y^{4}z^{4}-60000000000y^{2}z^{6}+7000000000000z^{8}}$ |
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 |
| 20.12.0.o.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.bp.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.ci.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.eo.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.er.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.jd.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.jh.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.js.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.jw.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.eh.1 | $40$ | $5$ | $5$ | $9$ | $5$ | $1^{6}\cdot2$ |
| 40.144.9.jb.1 | $40$ | $6$ | $6$ | $9$ | $1$ | $1^{6}\cdot2$ |
| 40.240.17.vd.1 | $40$ | $10$ | $10$ | $17$ | $8$ | $1^{12}\cdot2^{2}$ |
| 80.48.2.cx.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.cz.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.dv.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.dx.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.et.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.ev.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.fb.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.fd.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.48.1.bdp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bdx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bfl.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bft.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cin.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.civ.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cjs.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.cka.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.bsx.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.96.5.rj.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 240.48.2.fz.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.gb.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.gx.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.gz.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.hv.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.hx.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.id.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.if.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 280.48.1.bgb.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bgf.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bhh.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bhl.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bpx.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.bqb.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.brd.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.brh.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.13.kd.1 | $280$ | $8$ | $8$ | $13$ | $?$ | not computed |