Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $400$ | ||
| 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: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20C7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.7.763 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&2\\39&3\end{bmatrix}$, $\begin{bmatrix}21&36\\27&9\end{bmatrix}$, $\begin{bmatrix}23&4\\22&3\end{bmatrix}$, $\begin{bmatrix}29&8\\16&21\end{bmatrix}$, $\begin{bmatrix}33&14\\2&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.120.7.q.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{20}\cdot5^{12}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{7}$ |
| Newforms: | 20.2.a.a, 50.2.a.a, 50.2.a.b, 80.2.a.a, 400.2.a.c, 400.2.a.f, 400.2.a.h |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ 2 x^{2} + 2 x z - y^{2} + y z - t v $ |
| $=$ | $x y + 2 y z - z^{2} - w t - w v - t^{2} - t u - t v$ | |
| $=$ | $x y + x z + y^{2} - y z - w^{2} - w t - w u - w v$ | |
| $=$ | $x^{2} + y^{2} + y z - z^{2} + w t + w u + t^{2} + 2 t u + u^{2} + u v - v^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{10} - 136 x^{9} y + 321 x^{8} y^{2} - 9 x^{8} z^{2} + 200 x^{7} y^{3} + 122 x^{7} y z^{2} + \cdots + y^{2} z^{8} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=7$, 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.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -3x+y-2z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x+3y-z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x-2y-z$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{4}-3X^{3}Y-5X^{2}Y^{2}-9XY^{3}+2Y^{4}-X^{3}Z-7X^{2}YZ+7XY^{2}Z-15Y^{3}Z-6X^{2}Z^{2}+2XYZ^{2}+8Y^{2}Z^{2}-XZ^{3}+10YZ^{3}+3Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.120.7.q.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 16X^{10}-136X^{9}Y+321X^{8}Y^{2}-9X^{8}Z^{2}+200X^{7}Y^{3}+122X^{7}YZ^{2}-1348X^{6}Y^{4}-339X^{6}Y^{2}Z^{2}-X^{6}Z^{4}-8X^{5}Y^{5}-172X^{5}Y^{3}Z^{2}-28X^{5}YZ^{4}+2134X^{4}Y^{6}+1117X^{4}Y^{4}Z^{2}+159X^{4}Y^{2}Z^{4}+600X^{3}Y^{7}+414X^{3}Y^{5}Z^{2}+2X^{3}Y^{3}Z^{4}-4X^{3}YZ^{6}-692X^{2}Y^{8}-705X^{2}Y^{6}Z^{2}-314X^{2}Y^{4}Z^{4}+35X^{2}Y^{2}Z^{6}-144XY^{9}-140XY^{7}Z^{2}-16XY^{5}Z^{4}-20XY^{3}Z^{6}+81Y^{10}+160Y^{8}Z^{2}+78Y^{6}Z^{4}+Y^{2}Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.24.0-20.e.1.3 | $40$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 40.120.3-20.b.1.3 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.120.3-20.b.1.15 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
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.bg.1.7 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-20.bh.1.7 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-20.bo.1.5 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-20.bp.1.5 | $40$ | $2$ | $2$ | $13$ | $0$ | $1^{6}$ |
| 40.480.13-40.iq.1.2 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-40.ix.1.5 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-40.ku.1.7 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-40.lb.1.7 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.15-20.t.1.5 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-20.t.1.7 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-20.u.1.3 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-20.u.1.7 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-20.v.1.4 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-20.v.1.8 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-20.w.1.7 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-20.w.1.8 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-40.fo.1.1 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-40.fo.1.5 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-40.fp.1.1 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-40.fp.1.9 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-40.fq.1.3 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-40.fq.1.11 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-40.fr.1.5 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.480.15-40.fr.1.7 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 40.720.19-20.bx.1.8 | $40$ | $3$ | $3$ | $19$ | $1$ | $1^{12}$ |
| 120.480.13-60.ey.1.9 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.ez.1.5 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.fw.1.9 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.fx.1.5 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bim.1.10 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bit.1.11 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.boy.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bpf.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.15-60.br.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.br.1.19 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.bs.1.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.bs.1.21 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.bt.1.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.bt.1.21 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.bu.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.bu.1.17 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ki.1.4 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ki.1.12 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.kj.1.4 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.kj.1.20 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.kk.1.6 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.kk.1.22 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.kl.1.10 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.kl.1.14 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.13-140.ds.1.14 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.dt.1.10 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.ea.1.7 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.eb.1.7 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.zw.1.11 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.bad.1.8 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.bca.1.14 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.bch.1.14 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.15-140.bo.1.6 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.bo.1.14 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.bp.1.6 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.bp.1.14 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.bq.1.3 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.bq.1.7 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.br.1.3 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.br.1.7 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.hs.1.19 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.hs.1.23 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ht.1.19 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ht.1.27 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.hu.1.22 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.hu.1.30 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.hv.1.22 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.hv.1.30 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |