Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| 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: | 40.96.1.1124 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&24\\13&33\end{bmatrix}$, $\begin{bmatrix}11&16\\11&37\end{bmatrix}$, $\begin{bmatrix}23&0\\6&3\end{bmatrix}$, $\begin{bmatrix}27&16\\14&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.48.1.bc.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: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 4x $ |
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 |
|---|
| $(0:0:1)$, $(2:0:1)$, $(0:1:0)$, $(-2:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{159520x^{2}y^{12}z^{2}+100322304x^{2}y^{8}z^{6}-1196359680x^{2}y^{4}z^{10}+17175674880x^{2}z^{14}-688xy^{14}z+41328384xy^{10}z^{5}-1848377344xy^{6}z^{9}+21475885056xy^{2}z^{13}+y^{16}-12859648y^{12}z^{4}-292569088y^{8}z^{8}+4248305664y^{4}z^{12}+16777216z^{16}}{zy^{4}(52x^{2}y^{8}z+128256x^{2}y^{4}z^{5}+4177920x^{2}z^{9}+xy^{10}+17152xy^{6}z^{4}+3149824xy^{2}z^{8}+1120y^{8}z^{3}+534528y^{4}z^{7}+65536z^{11})}$ |
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.q.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-8.q.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-8.t.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.0-8.t.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.48.1-8.n.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1-8.n.1.4 | $40$ | $2$ | $2$ | $1$ | $0$ | 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.m.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-8.m.1.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.ck.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.ck.1.5 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.480.17-40.ff.1.6 | $40$ | $5$ | $5$ | $17$ | $5$ | $1^{14}\cdot2$ |
| 40.576.17-40.mu.1.4 | $40$ | $6$ | $6$ | $17$ | $2$ | $1^{14}\cdot2$ |
| 40.960.33-40.xh.1.5 | $40$ | $10$ | $10$ | $33$ | $10$ | $1^{28}\cdot2^{2}$ |
| 80.192.3-16.cl.1.4 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cl.1.6 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.co.1.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cq.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cq.1.5 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cr.1.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cr.2.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.ct.1.4 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cv.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-16.cv.1.6 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gq.1.9 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gq.1.20 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gt.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gv.1.8 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gv.1.19 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gw.1.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gw.2.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.gx.1.10 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.hj.1.3 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.3-80.hj.1.20 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.192.5-16.bt.1.3 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-16.bt.2.3 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-16.bu.1.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-16.bu.1.7 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.ef.1.11 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.ef.2.11 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.eg.1.3 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5-80.eg.1.23 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.1-24.ct.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.ct.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.qj.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.qj.1.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.288.9-24.uv.1.11 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.384.9-24.hx.1.11 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.192.3-48.fo.1.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fo.1.19 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fr.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.ft.1.6 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.ft.1.11 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fu.1.3 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fu.2.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.fv.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.gh.1.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-48.gh.1.18 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.qy.1.14 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.qy.1.22 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rf.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rh.1.13 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rh.1.38 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ri.1.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ri.2.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.rj.1.11 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.st.1.12 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.st.1.20 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.5-48.eb.1.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-48.eb.2.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-48.ec.1.13 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-48.ec.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.ld.1.19 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.ld.2.11 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.le.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.le.1.44 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.192.1-56.ck.1.3 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-56.ck.1.8 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.po.1.7 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.po.1.10 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |