Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $200$ | ||
| Index: | $120$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $4 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $5^{2}\cdot10\cdot40$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40B4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.120.4.59 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&23\\4&9\end{bmatrix}$, $\begin{bmatrix}3&24\\32&27\end{bmatrix}$, $\begin{bmatrix}9&35\\32&11\end{bmatrix}$, $\begin{bmatrix}31&6\\12&9\end{bmatrix}$, $\begin{bmatrix}39&15\\12&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.60.4.bl.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $48$ |
| Full 40-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{6}\cdot5^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 50.2.a.b$^{3}$, 200.2.a.e |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ x^{2} - x z + y w + 2 z^{2} $ |
| $=$ | $2 x^{2} z + x y w + 2 x w^{2} - y^{2} z - y z w - 2 z^{3} + 2 z w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{6} - 2 x^{4} y^{2} + 9 x^{4} y z + 5 x^{4} z^{2} + 2 x^{2} y^{4} + 3 x^{2} y^{2} z^{2} + \cdots + y^{3} z^{3} $ |
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.
| Canonical model |
|---|
| $(0:1:0:0)$, $(0:0:0:1)$, $(1:-2:1:1)$, $(-1:-2:-1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{247959xyz^{7}w-1069305xyz^{5}w^{3}+285486xyz^{3}w^{5}+106620xyzw^{7}-362824xz^{9}-2275690xz^{7}w^{2}+1375404xz^{5}w^{4}-604018xz^{3}w^{6}+223160xzw^{8}+128y^{10}+1280y^{9}w+5760y^{8}w^{2}+12800y^{7}w^{3}+8960y^{6}w^{4}-23040y^{5}w^{5}-75520y^{4}w^{6}-102400y^{3}w^{7}-67200y^{2}w^{8}-1499362yz^{8}w+1140415yz^{6}w^{3}+19449yz^{4}w^{5}-23530yz^{2}w^{7}-17540yw^{9}-377232z^{10}+1996906z^{8}w^{2}-2345416z^{6}w^{4}+633222z^{4}w^{6}+152680z^{2}w^{8}+32w^{10}}{8xyz^{7}w-15xyz^{5}w^{3}+4xyz^{3}w^{5}+xyzw^{7}+32xz^{9}-40xz^{7}w^{2}+18xz^{5}w^{4}+2xz^{3}w^{6}-6xzw^{8}+16yz^{8}w+10yz^{6}w^{3}-25yz^{4}w^{5}+10yz^{2}w^{7}+yw^{9}+32z^{8}w^{2}-42z^{6}w^{4}+26z^{4}w^{6}-2z^{2}w^{8}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.60.4.bl.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{6}-2X^{4}Y^{2}+9X^{4}YZ+5X^{4}Z^{2}+2X^{2}Y^{4}+3X^{2}Y^{2}Z^{2}+5X^{2}YZ^{3}+2X^{2}Z^{4}+Y^{5}Z+2Y^{4}Z^{2}+Y^{3}Z^{3} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 8.24.0-8.n.1.1 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-8.n.1.1 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.240.8-40.v.1.1 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-40.z.1.3 | $40$ | $2$ | $2$ | $8$ | $2$ | $1^{4}$ |
| 40.240.8-40.bj.1.1 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-40.bk.1.3 | $40$ | $2$ | $2$ | $8$ | $2$ | $1^{4}$ |
| 40.240.8-40.cj.1.1 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-40.cl.1.4 | $40$ | $2$ | $2$ | $8$ | $2$ | $1^{4}$ |
| 40.240.8-40.cn.1.4 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-40.cp.1.4 | $40$ | $2$ | $2$ | $8$ | $2$ | $1^{4}$ |
| 40.240.8-40.da.1.3 | $40$ | $2$ | $2$ | $8$ | $2$ | $2^{2}$ |
| 40.240.8-40.da.1.14 | $40$ | $2$ | $2$ | $8$ | $2$ | $2^{2}$ |
| 40.240.8-40.da.2.7 | $40$ | $2$ | $2$ | $8$ | $2$ | $2^{2}$ |
| 40.240.8-40.da.2.10 | $40$ | $2$ | $2$ | $8$ | $2$ | $2^{2}$ |
| 40.240.8-40.db.1.2 | $40$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 40.240.8-40.db.1.15 | $40$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 40.240.8-40.db.2.6 | $40$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 40.240.8-40.db.2.11 | $40$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 40.240.8-40.dc.1.4 | $40$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 40.240.8-40.dc.1.13 | $40$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 40.240.8-40.dc.2.2 | $40$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 40.240.8-40.dc.2.15 | $40$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 40.240.8-40.dd.1.4 | $40$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 40.240.8-40.dd.1.13 | $40$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 40.240.8-40.dd.2.5 | $40$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 40.240.8-40.dd.2.12 | $40$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 40.360.10-40.cj.1.1 | $40$ | $3$ | $3$ | $10$ | $0$ | $1^{6}$ |
| 40.480.13-40.of.1.2 | $40$ | $4$ | $4$ | $13$ | $1$ | $1^{9}$ |
| 80.240.8-80.q.1.1 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.q.1.32 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.q.2.13 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.q.2.20 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.r.1.2 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.r.1.31 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.r.2.14 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.r.2.19 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.s.1.6 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.s.1.27 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.s.2.10 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.s.2.23 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.t.1.5 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.t.1.28 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.t.2.9 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.t.2.24 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.u.1.1 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.u.1.32 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.v.1.1 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.v.1.32 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.y.1.1 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.y.1.32 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.z.1.1 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.z.1.32 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.9-80.a.1.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.240.9-80.a.1.32 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.240.9-80.b.1.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.240.9-80.b.1.32 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.240.9-80.e.1.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.240.9-80.e.1.32 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.240.9-80.f.1.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.240.9-80.f.1.32 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.240.8-120.db.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.dd.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.df.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.dh.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fb.1.20 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fd.1.22 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ff.1.12 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fh.1.22 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gg.1.5 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gg.1.28 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gg.2.7 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gg.2.26 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gh.1.15 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gh.1.18 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gh.2.13 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gh.2.20 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gi.1.13 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gi.1.20 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gi.2.15 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gi.2.18 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gj.1.7 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gj.1.26 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gj.2.5 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gj.2.28 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.360.14-120.fd.1.7 | $120$ | $3$ | $3$ | $14$ | $?$ | not computed |
| 120.480.17-120.brb.1.31 | $120$ | $4$ | $4$ | $17$ | $?$ | not computed |
| 240.240.8-240.q.1.32 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.q.1.33 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.q.2.20 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.q.2.45 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.r.1.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.r.1.64 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.r.2.13 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.r.2.52 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.s.1.5 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.s.1.60 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.s.2.5 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.s.2.60 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.t.1.28 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.t.1.37 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.t.2.28 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.t.2.37 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.u.1.5 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.u.1.60 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.v.1.13 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.v.1.52 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.y.1.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.y.1.58 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.z.1.15 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.z.1.50 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.9-240.a.1.21 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.240.9-240.a.1.44 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.240.9-240.b.1.22 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.240.9-240.b.1.43 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.240.9-240.e.1.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.240.9-240.e.1.55 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.240.9-240.f.1.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.240.9-240.f.1.59 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.240.8-280.dr.1.1 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.dt.1.5 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.dv.1.1 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.dx.1.5 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.ex.1.9 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.ez.1.5 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.fb.1.9 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.fd.1.5 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.gc.1.2 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.gc.1.31 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.gc.2.6 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.gc.2.27 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.gd.1.14 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.gd.1.19 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.gd.2.10 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.gd.2.23 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.ge.1.10 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.ge.1.23 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.ge.2.14 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.ge.2.19 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.gf.1.6 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.gf.1.27 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.gf.2.2 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.gf.2.31 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |