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: | $1$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20C7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.7.105 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}15&18\\28&25\end{bmatrix}$, $\begin{bmatrix}23&8\\2&25\end{bmatrix}$, $\begin{bmatrix}23&32\\18&21\end{bmatrix}$, $\begin{bmatrix}25&14\\26&31\end{bmatrix}$, $\begin{bmatrix}39&36\\22&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.7.c.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: | $2^{28}\cdot5^{12}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 50.2.a.b$^{2}$, 100.2.a.a, 320.2.a.c, 320.2.a.f, 1600.2.a.j, 1600.2.a.x |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ 4 x y - 2 x z - 2 y z + 2 z^{2} + t v $ |
| $=$ | $2 y^{2} - 4 z^{2} + w u - t^{2} + u^{2}$ | |
| $=$ | $2 x^{2} - 2 y z + 2 z^{2} - w t - w u + 2 t^{2} - t v - u^{2} - u v$ | |
| $=$ | $x t + x u - x v - y w + 2 y t - y u - y v - z u - z v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{12} + 224 x^{11} y + 1000 x^{10} y^{2} - 32 x^{10} z^{2} + 1496 x^{9} y^{3} - 600 x^{9} y z^{2} + \cdots + 36 y^{4} z^{8} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=3$, 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 :
| $\displaystyle X$ | $=$ | $\displaystyle -2x-y+3z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4x+2y-z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x+3y+z$ |
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.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 16X^{12}+224X^{11}Y+1000X^{10}Y^{2}-32X^{10}Z^{2}+1496X^{9}Y^{3}-600X^{9}YZ^{2}+609X^{8}Y^{4}-3208X^{8}Y^{2}Z^{2}+36X^{8}Z^{4}-164X^{7}Y^{5}-5010X^{7}Y^{3}Z^{2}+548X^{7}YZ^{4}-50X^{6}Y^{6}-1994X^{6}Y^{4}Z^{2}+2960X^{6}Y^{2}Z^{4}+4X^{5}Y^{7}+602X^{5}Y^{5}Z^{2}+4628X^{5}Y^{3}Z^{4}-252X^{5}YZ^{6}+X^{4}Y^{8}+182X^{4}Y^{6}Z^{2}+1828X^{4}Y^{4}Z^{4}-878X^{4}Y^{2}Z^{6}-16X^{3}Y^{7}Z^{2}-572X^{3}Y^{5}Z^{4}-1236X^{3}Y^{3}Z^{6}-4X^{2}Y^{8}Z^{2}-172X^{2}Y^{6}Z^{4}-578X^{2}Y^{4}Z^{6}+36X^{2}Y^{2}Z^{8}+16XY^{7}Z^{4}+48XY^{5}Z^{6}+72XY^{3}Z^{8}+4Y^{8}Z^{4}+16Y^{6}Z^{6}+36Y^{4}Z^{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.1.6 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.24.0-40.a.1.7 | $40$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 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.1.5 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.13-40.o.1.8 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-40.s.1.6 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
| 40.480.13-40.u.1.5 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-40.bk.1.6 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 40.480.13-40.bm.1.4 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-40.bq.1.7 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
| 40.480.13-40.bs.1.8 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.15-40.d.1.4 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.e.1.4 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.e.1.6 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.h.1.8 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.h.1.10 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.j.1.4 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
| 40.480.15-40.j.1.8 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
| 40.480.15-40.bp.1.12 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.bp.1.16 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.bq.1.12 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.bq.1.14 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.bs.1.12 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.bs.1.14 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.bt.1.12 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.bt.1.14 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.720.19-40.bl.1.9 | $40$ | $3$ | $3$ | $19$ | $4$ | $1^{12}$ |
| 120.480.13-120.bw.1.9 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.by.1.16 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.cc.1.9 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.ce.1.9 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.eq.1.5 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.es.1.4 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.ew.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.ey.1.16 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.15-120.o.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.o.1.24 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.q.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.q.1.24 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.u.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.u.1.24 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.w.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.w.1.24 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dd.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dd.1.24 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.de.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.de.1.24 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dg.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dg.1.24 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dh.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dh.1.24 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.13-280.ci.1.9 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.ck.1.15 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.co.1.12 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.cq.1.9 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.dg.1.11 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.di.1.9 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.dm.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.do.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.15-280.p.1.15 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.p.1.21 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.q.1.11 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.q.1.29 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.s.1.5 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.s.1.31 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.t.1.15 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.t.1.27 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cf.1.14 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cf.1.28 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cg.1.10 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cg.1.32 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ci.1.14 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ci.1.28 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cj.1.16 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cj.1.26 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |