Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $200$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $7 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $10^{4}\cdot20^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20B7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.7.708 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}13&38\\20&37\end{bmatrix}$, $\begin{bmatrix}15&32\\12&5\end{bmatrix}$, $\begin{bmatrix}27&6\\14&23\end{bmatrix}$, $\begin{bmatrix}31&22\\16&9\end{bmatrix}$, $\begin{bmatrix}31&22\\18&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.120.7.e.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{14}\cdot5^{14}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 50.2.a.b$^{3}$, 100.2.a.a, 200.2.a.b, 200.2.a.d, 200.2.a.e |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x w + x t + x u + x v - z u $ |
| $=$ | $x w - x t + x u + y u - z u$ | |
| $=$ | $y w + y t + y u + y v - 2 z t - z v$ | |
| $=$ | $x^{2} + 3 x y + x z + t u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2500 x^{12} - 7000 x^{10} y^{2} - 3500 x^{10} y z + 875 x^{10} z^{2} + 4900 x^{8} y^{4} + \cdots + y^{2} z^{10} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 10.60.3.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle 2x+y-3z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -4x-2y+z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x+2y-z$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}-3X^{3}Y-5X^{2}Y^{2}-4XY^{3}-2Y^{4}+3X^{3}Z-18X^{2}YZ-17XY^{2}Z+4Y^{3}Z-5X^{2}Z^{2}+17XYZ^{2}-6Y^{2}Z^{2}+4XZ^{3}+4YZ^{3}-2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.120.7.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ 2500X^{12}-7000X^{10}Y^{2}+4900X^{8}Y^{4}-3500X^{10}YZ+4900X^{8}Y^{3}Z+875X^{10}Z^{2}-4825X^{8}Y^{2}Z^{2}+10400X^{6}Y^{4}Z^{2}-3025X^{8}YZ^{3}+10400X^{6}Y^{3}Z^{3}-225X^{8}Z^{4}+2250X^{6}Y^{2}Z^{4}+2200X^{4}Y^{4}Z^{4}-175X^{6}YZ^{5}+2200X^{4}Y^{3}Z^{5}-75X^{6}Z^{6}+660X^{4}Y^{2}Z^{6}+160X^{2}Y^{4}Z^{6}+55X^{4}YZ^{7}+160X^{2}Y^{3}Z^{7}-5X^{4}Z^{8}+50X^{2}Y^{2}Z^{8}+4Y^{4}Z^{8}+5X^{2}YZ^{9}+4Y^{3}Z^{9}+Y^{2}Z^{10} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.120.3-10.a.1.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.120.3-10.a.1.5 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.480.13-20.f.1.2 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 40.480.13-20.g.1.2 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-20.j.1.1 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 40.480.13-20.k.1.1 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.13-40.r.1.5 | $40$ | $2$ | $2$ | $13$ | $6$ | $1^{6}$ |
| 40.480.13-40.u.1.5 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-40.bd.1.2 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 40.480.13-40.bg.1.2 | $40$ | $2$ | $2$ | $13$ | $7$ | $1^{6}$ |
| 40.480.15-20.f.1.2 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-20.h.1.5 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-20.h.1.8 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-20.l.1.2 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-20.l.1.3 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-20.o.1.2 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-20.o.1.4 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.bb.1.3 | $40$ | $2$ | $2$ | $15$ | $7$ | $1^{8}$ |
| 40.480.15-40.bb.1.7 | $40$ | $2$ | $2$ | $15$ | $7$ | $1^{8}$ |
| 40.480.15-40.bf.1.3 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.bf.1.7 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.bp.1.2 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.bp.1.6 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.by.1.2 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.by.1.6 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.720.19-20.s.1.2 | $40$ | $3$ | $3$ | $19$ | $2$ | $1^{12}$ |
| 120.480.13-60.v.1.5 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.w.1.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.bh.1.5 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.bi.1.4 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.cn.1.4 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.cq.1.4 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.dx.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.ea.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.15-60.n.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.n.1.11 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.p.1.6 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.p.1.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.bc.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.bc.1.8 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.bf.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.bf.1.11 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bz.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bz.1.27 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cf.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cf.1.27 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dp.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dp.1.27 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dy.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dy.1.27 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.13-140.bk.1.3 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.bm.1.2 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.bo.1.2 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.bq.1.2 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.eg.1.11 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.em.1.10 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.es.1.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.ey.1.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.15-140.u.1.7 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.u.1.11 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.v.1.3 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.v.1.15 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.bc.1.1 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.bc.1.11 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.bd.1.3 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.bd.1.9 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.da.1.3 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.da.1.31 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dd.1.3 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dd.1.31 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dy.1.3 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dy.1.21 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.eb.1.3 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.eb.1.13 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |