Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $1600$ | ||
| Index: | $96$ | $\PSL_2$-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: | $0$ | ||||||
| $\Q$-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: | 40.96.1.234 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&8\\8&37\end{bmatrix}$, $\begin{bmatrix}13&8\\33&35\end{bmatrix}$, $\begin{bmatrix}21&32\\28&37\end{bmatrix}$, $\begin{bmatrix}23&24\\37&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.1.cy.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $7680$ |
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} - 100x $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(10:0:1)$, $(-10:0:1)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{5^2}\cdot\frac{99700000x^{2}y^{12}z^{2}+979710000000000x^{2}y^{8}z^{6}-182550000000000000000x^{2}y^{4}z^{10}+40950000000000000000000000x^{2}z^{14}-17200xy^{14}z+16143900000000xy^{10}z^{5}-11281600000000000000xy^{6}z^{9}+2048100000000000000000000xy^{2}z^{13}+y^{16}-200932000000y^{12}z^{4}-71428000000000000y^{8}z^{8}+16206000000000000000000y^{4}z^{12}+1000000000000000000000000z^{16}}{zy^{4}(1300x^{2}y^{8}z+50100000000x^{2}y^{4}z^{5}+25500000000000000x^{2}z^{9}+xy^{10}+268000000xy^{6}z^{4}+769000000000000xy^{2}z^{8}+700000y^{8}z^{3}+5220000000000y^{4}z^{7}+10000000000000000z^{11})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.q.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-8.q.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-40.z.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-40.z.1.8 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.1-40.n.1.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1-40.n.1.8 | $40$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.192.1-40.cm.1.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.cm.1.7 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.cm.2.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.cm.2.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.480.17-40.fh.1.5 | $40$ | $5$ | $5$ | $17$ | $5$ | $1^{14}\cdot2$ |
| 40.576.17-40.mw.1.4 | $40$ | $6$ | $6$ | $17$ | $4$ | $1^{14}\cdot2$ |
| 40.960.33-40.xj.1.5 | $40$ | $10$ | $10$ | $33$ | $8$ | $1^{28}\cdot2^{2}$ |
| 80.192.3-80.gl.1.7 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gl.1.11 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gp.1.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gp.1.12 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gz.1.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gz.1.9 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.hf.1.7 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.hf.1.13 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.hi.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.hi.1.12 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.hl.1.7 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.hl.1.10 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.1-120.qn.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.qn.1.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.qn.2.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.qn.2.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.288.9-120.cjd.1.23 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.384.9-120.wr.1.23 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.192.3-240.qq.1.16 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.qq.1.21 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rb.1.12 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rb.1.23 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ru.1.9 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ru.1.19 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rz.1.13 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rz.1.27 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sw.1.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.sw.1.22 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.tf.1.16 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.tf.1.18 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.192.1-280.pq.1.8 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.pq.1.9 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.pq.2.6 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.pq.2.11 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |