Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $5 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $10^{12}$ | Cusp orbits | $4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10B5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.5.84 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&8\\32&19\end{bmatrix}$, $\begin{bmatrix}7&34\\16&23\end{bmatrix}$, $\begin{bmatrix}13&2\\14&37\end{bmatrix}$, $\begin{bmatrix}19&26\\34&23\end{bmatrix}$, $\begin{bmatrix}21&32\\6&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.5.b.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: | $2^{16}\cdot5^{10}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}$ |
| Newforms: | 50.2.a.b$^{2}$, 100.2.a.a, 1600.2.a.j$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 8 x^{2} - 8 x y + 14 x z - 8 y^{2} + 8 y z + 8 z^{2} - w^{2} + w t $ |
| $=$ | $6 x^{2} - 16 x y - 2 x z + 24 y^{2} + 16 y z + 6 z^{2} + w^{2} - w t + t^{2}$ | |
| $=$ | $16 x^{2} - 16 x y + 8 x z - 16 y^{2} - 24 y z - 14 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y^{4} + 20 x^{4} y^{2} z^{2} + 20 x^{4} z^{4} + 4 x^{3} y^{5} + 80 x^{3} y^{3} z^{2} + \cdots + 3600 z^{8} $ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=13$, and therefore no 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 :
| $\displaystyle X$ | $=$ | $\displaystyle x-3y-z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 3x+y+2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2x-y+3z$ |
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 40.120.5.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{20}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y^{4}+20X^{4}Y^{2}Z^{2}+20X^{4}Z^{4}+4X^{3}Y^{5}+80X^{3}Y^{3}Z^{2}+80X^{3}YZ^{4}+2X^{2}Y^{6}+20X^{2}Y^{4}Z^{2}-440X^{2}Y^{2}Z^{4}+400X^{2}Z^{6}-4XY^{7}-120XY^{5}Z^{2}-1040XY^{3}Z^{4}+800XYZ^{6}+Y^{8}+40Y^{6}Z^{2}+600Y^{4}Z^{4}+4000Y^{2}Z^{6}+3600Z^{8} $ |
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.1.1 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.120.3-10.a.1.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
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.h.1.2 | $40$ | $2$ | $2$ | $13$ | $6$ | $1^{8}$ |
| 40.480.13-40.h.1.3 | $40$ | $2$ | $2$ | $13$ | $6$ | $1^{8}$ |
| 40.480.13-40.i.1.3 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{8}$ |
| 40.480.13-40.i.1.6 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{8}$ |
| 40.480.13-40.i.1.7 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{8}$ |
| 40.480.13-40.k.1.6 | $40$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 40.480.13-40.k.1.10 | $40$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 40.480.13-40.k.1.20 | $40$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 40.480.13-40.l.1.2 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
| 40.480.13-40.l.1.3 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
| 40.480.13-40.l.1.8 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
| 40.480.13-40.t.1.1 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{8}$ |
| 40.480.13-40.t.1.6 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{8}$ |
| 40.480.13-40.t.1.8 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{8}$ |
| 40.480.13-40.u.1.1 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 40.480.13-40.u.1.5 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 40.480.13-40.u.1.8 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 40.480.13-40.w.1.1 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 40.480.13-40.w.1.5 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 40.480.13-40.w.1.6 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 40.480.13-40.x.1.1 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
| 40.480.13-40.x.1.7 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
| 40.480.13-40.x.1.8 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
| 40.720.13-40.d.1.1 | $40$ | $3$ | $3$ | $13$ | $3$ | $1^{8}$ |
| 120.480.13-120.h.1.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.h.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.h.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.i.1.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.i.1.13 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.i.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.k.1.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.k.1.11 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.k.1.16 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.l.1.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.l.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.l.1.16 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.br.1.3 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.br.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.br.1.16 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bs.1.3 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bs.1.6 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bs.1.16 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bu.1.3 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bu.1.6 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bu.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bv.1.3 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bv.1.10 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bv.1.11 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.h.1.4 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.h.1.9 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.h.1.12 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.i.1.4 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.i.1.13 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.i.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.k.1.4 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.k.1.7 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.k.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.l.1.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.l.1.4 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.l.1.14 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.t.1.2 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.t.1.12 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.t.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.u.1.2 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.u.1.11 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.u.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.w.1.2 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.w.1.9 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.w.1.10 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.x.1.2 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.x.1.13 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.x.1.14 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |