Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
| 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: | $2$ | ||||||
| $\Q$-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: | 40.240.7.198 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&24\\32&23\end{bmatrix}$, $\begin{bmatrix}9&10\\36&11\end{bmatrix}$, $\begin{bmatrix}33&4\\26&19\end{bmatrix}$, $\begin{bmatrix}33&24\\36&29\end{bmatrix}$, $\begin{bmatrix}37&8\\2&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.7.f.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: | $2^{28}\cdot5^{14}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 50.2.a.b$^{2}$, 100.2.a.a, 1600.2.a.a, 1600.2.a.f, 1600.2.a.q, 1600.2.a.v |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x w + x t - x v + z u $ |
| $=$ | $x w - x t - x u - y u + z u$ | |
| $=$ | $y w + y t - y v - 2 z t - z u + z v$ | |
| $=$ | $2 x^{2} + 6 x y + 2 x z - t u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2500 x^{12} - 3500 x^{10} y^{2} + 3500 x^{10} y z + 1750 x^{10} z^{2} + 1225 x^{8} y^{4} + \cdots + 16 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 :
| $\displaystyle X$ | $=$ | $\displaystyle 2x+y-3z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -4x-2y+z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x-3y-z$ |
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 40.120.7.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}u$ |
Equation of the image curve:
| $0$ | $=$ | $ 2500X^{12}-3500X^{10}Y^{2}+1225X^{8}Y^{4}+3500X^{10}YZ-2450X^{8}Y^{3}Z+1750X^{10}Z^{2}-4825X^{8}Y^{2}Z^{2}+5200X^{6}Y^{4}Z^{2}+6050X^{8}YZ^{3}-10400X^{6}Y^{3}Z^{3}-900X^{8}Z^{4}+4500X^{6}Y^{2}Z^{4}+2200X^{4}Y^{4}Z^{4}+700X^{6}YZ^{5}-4400X^{4}Y^{3}Z^{5}-600X^{6}Z^{6}+2640X^{4}Y^{2}Z^{6}+320X^{2}Y^{4}Z^{6}-440X^{4}YZ^{7}-640X^{2}Y^{3}Z^{7}-80X^{4}Z^{8}+400X^{2}Y^{2}Z^{8}+16Y^{4}Z^{8}-80X^{2}YZ^{9}-32Y^{3}Z^{9}+16Y^{2}Z^{10} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.120.3-10.a.1.2 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.120.3-10.a.1.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.480.13-40.m.1.7 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.13-40.n.1.8 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
| 40.480.13-40.v.1.1 | $40$ | $2$ | $2$ | $13$ | $7$ | $1^{6}$ |
| 40.480.13-40.w.1.1 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-40.y.1.2 | $40$ | $2$ | $2$ | $13$ | $6$ | $1^{6}$ |
| 40.480.13-40.z.1.1 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-40.bh.1.8 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-40.bi.1.19 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.15-40.bc.1.3 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.bc.1.6 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.bd.1.3 | $40$ | $2$ | $2$ | $15$ | $7$ | $1^{8}$ |
| 40.480.15-40.bd.1.13 | $40$ | $2$ | $2$ | $15$ | $7$ | $1^{8}$ |
| 40.480.15-40.be.1.8 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.be.1.13 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.bf.1.7 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.bf.1.13 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.br.1.7 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.br.1.14 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.bt.1.7 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.bt.1.14 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.bu.1.5 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.bu.1.15 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.bw.1.1 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.bw.1.14 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.720.19-40.bz.1.11 | $40$ | $3$ | $3$ | $19$ | $5$ | $1^{12}$ |
| 120.480.13-120.ci.1.5 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.cj.1.15 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.cr.1.13 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.cs.1.11 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.ds.1.9 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.dt.1.5 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.eb.1.15 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.ec.1.15 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.15-120.ca.1.28 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ca.1.29 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cb.1.29 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cb.1.32 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cg.1.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cg.1.29 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ch.1.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ch.1.28 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dr.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dr.1.31 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dt.1.5 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dt.1.26 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.du.1.26 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.du.1.31 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dw.1.30 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dw.1.31 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.13-280.eh.1.13 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.ei.1.15 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.en.1.15 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.eo.1.4 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.et.1.15 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.eu.1.10 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.ez.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.fa.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.15-280.de.1.22 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.de.1.23 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.df.1.22 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.df.1.23 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dh.1.7 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dh.1.22 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.di.1.7 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.di.1.22 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ec.1.11 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ec.1.28 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ed.1.11 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ed.1.28 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ef.1.29 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ef.1.30 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.eg.1.29 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.eg.1.30 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |