Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $800$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $1^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.1.153 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}27&8\\22&39\end{bmatrix}$, $\begin{bmatrix}31&8\\31&37\end{bmatrix}$, $\begin{bmatrix}33&16\\27&3\end{bmatrix}$, $\begin{bmatrix}35&16\\19&17\end{bmatrix}$, $\begin{bmatrix}37&32\\6&37\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 40.96.1-40.cx.1.1, 40.96.1-40.cx.1.2, 40.96.1-40.cx.1.3, 40.96.1-40.cx.1.4, 40.96.1-40.cx.1.5, 40.96.1-40.cx.1.6, 80.96.1-40.cx.1.1, 80.96.1-40.cx.1.2, 80.96.1-40.cx.1.3, 80.96.1-40.cx.1.4, 80.96.1-40.cx.1.5, 80.96.1-40.cx.1.6, 120.96.1-40.cx.1.1, 120.96.1-40.cx.1.2, 120.96.1-40.cx.1.3, 120.96.1-40.cx.1.4, 120.96.1-40.cx.1.5, 120.96.1-40.cx.1.6, 240.96.1-40.cx.1.1, 240.96.1-40.cx.1.2, 240.96.1-40.cx.1.3, 240.96.1-40.cx.1.4, 240.96.1-40.cx.1.5, 240.96.1-40.cx.1.6, 280.96.1-40.cx.1.1, 280.96.1-40.cx.1.2, 280.96.1-40.cx.1.3, 280.96.1-40.cx.1.4, 280.96.1-40.cx.1.5, 280.96.1-40.cx.1.6 |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $15360$ |
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} - 25x $ |
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 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^2}{5^2}\cdot\frac{6231250x^{2}y^{12}z^{2}+956748046875x^{2}y^{8}z^{6}-2785491943359375x^{2}y^{4}z^{10}+9763240814208984375x^{2}z^{14}-4300xy^{14}z+63062109375xy^{10}z^{5}-688574218750000xy^{6}z^{9}+1953220367431640625xy^{2}z^{13}+y^{16}-3139562500y^{12}z^{4}-17438476562500y^{8}z^{8}+61820983886718750y^{4}z^{12}+59604644775390625z^{16}}{zy^{4}(325x^{2}y^{8}z+195703125x^{2}y^{4}z^{5}+1556396484375x^{2}z^{9}+xy^{10}+4187500xy^{6}z^{4}+187744140625xy^{2}z^{8}+43750y^{8}z^{3}+5097656250y^{4}z^{7}+152587890625z^{11})}$ |
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 |
|---|---|---|---|---|---|---|
| 8.24.0.q.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 20.24.0.d.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.dg.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0.dh.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.1.m.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.24.1.da.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.24.1.db.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 |
|---|---|---|---|---|---|---|
| 40.96.1.cl.1 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.96.1.cl.2 | $40$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 40.240.17.fg.1 | $40$ | $5$ | $5$ | $17$ | $4$ | $1^{14}\cdot2$ |
| 40.288.17.mv.1 | $40$ | $6$ | $6$ | $17$ | $4$ | $1^{14}\cdot2$ |
| 40.480.33.xi.1 | $40$ | $10$ | $10$ | $33$ | $6$ | $1^{28}\cdot2^{2}$ |
| 80.96.3.gk.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.go.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.gy.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.hc.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.hh.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.hk.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.1.qm.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1.qm.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.144.9.cjc.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.9.wq.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.96.3.qn.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.ra.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.rs.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.rw.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.sv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.tc.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.96.1.pp.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.96.1.pp.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |