Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| 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 | $1^{4}\cdot2^{2}$ | ||
| 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: | 8G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.1.1035 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&4\\18&21\end{bmatrix}$, $\begin{bmatrix}17&32\\8&5\end{bmatrix}$, $\begin{bmatrix}25&8\\26&19\end{bmatrix}$, $\begin{bmatrix}31&32\\22&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.48.1.k.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $7680$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 11x + 14 $ |
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 |
|---|
| $(2:0:1)$, $(0:1:0)$, $(1:-2:1)$, $(1:2:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{24x^{2}y^{14}-424786x^{2}y^{12}z^{2}+1011536664x^{2}y^{10}z^{4}-588578534409x^{2}y^{8}z^{6}+109936935701376x^{2}y^{6}z^{8}-8365764142695129x^{2}y^{4}z^{10}+273771076691951604x^{2}y^{2}z^{12}-3203095830928031745x^{2}z^{14}-1036xy^{14}z+6584712xy^{12}z^{3}-10157502399xy^{10}z^{5}+4099344479562xy^{8}z^{7}-605868487863256xy^{6}z^{9}+39304781176502856xy^{4}z^{11}-1145262787644096537xy^{2}z^{13}+12262818962286313470xz^{15}-y^{16}+20112y^{14}z^{2}-86989668y^{12}z^{4}+78616007544y^{10}z^{6}-20141247406412y^{8}z^{8}+2007992773878816y^{6}z^{10}-89258362532785194y^{4}z^{12}+1745005629946200144y^{2}z^{14}-11713254600860499961z^{16}}{zy^{4}(287x^{2}y^{8}z-66800x^{2}y^{6}z^{3}+2293821x^{2}y^{4}z^{5}-4x^{2}y^{2}z^{7}+x^{2}z^{9}+xy^{10}-2390xy^{8}z^{2}+325308xy^{6}z^{4}-8781712xy^{4}z^{6}-7xy^{2}z^{8}+2xz^{10}-24y^{10}z+13046y^{8}z^{3}-776976y^{6}z^{5}+8388142y^{4}z^{7}+32y^{2}z^{9}-7z^{11})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.48.0-8.e.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-8.e.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.1-8.c.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1-8.c.1.11 | $40$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.192.1-8.g.1.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-8.h.1.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 80.192.3-16.m.2.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.n.2.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.bc.2.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.bd.2.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.5-16.t.2.12 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-16.u.2.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-16.v.2.10 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-16.w.2.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.1-24.y.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.z.2.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.288.9-24.dn.2.4 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.384.9-24.by.2.12 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 40.192.1-40.y.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.z.1.5 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.480.17-40.bf.2.3 | $40$ | $5$ | $5$ | $17$ | $1$ | $1^{6}\cdot2^{5}$ |
| 40.576.17-40.ci.1.7 | $40$ | $6$ | $6$ | $17$ | $1$ | $1^{6}\cdot2\cdot4^{2}$ |
| 40.960.33-40.fp.2.1 | $40$ | $10$ | $10$ | $33$ | $3$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 240.192.3-48.be.2.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.bf.2.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.by.2.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.bz.2.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.5-48.ch.2.20 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-48.ci.2.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-48.cj.2.20 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-48.ck.2.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.192.1-56.y.1.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-56.z.2.5 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.3-80.bs.2.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.bt.2.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.co.2.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.cp.2.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.5-80.ch.2.14 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.ci.2.3 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.cj.2.16 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.ck.2.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.1-120.da.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.db.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.3-240.do.2.12 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.dp.2.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ek.2.12 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.el.2.14 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.5-240.gx.2.37 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.gy.2.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.gz.2.38 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.ha.2.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.192.1-280.da.1.14 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.db.1.12 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |