Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $200$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20J3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.3.990 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&20\\20&33\end{bmatrix}$, $\begin{bmatrix}21&4\\10&31\end{bmatrix}$, $\begin{bmatrix}37&18\\10&39\end{bmatrix}$, $\begin{bmatrix}39&4\\0&11\end{bmatrix}$, $\begin{bmatrix}39&26\\0&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.72.3.d.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $64$ |
| Full 40-torsion field degree: | $5120$ |
Jacobian
| Conductor: | $2^{7}\cdot5^{5}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}$ |
| Newforms: | 20.2.a.a, 100.2.a.a, 200.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x y w + x w t - z w t $ |
| $=$ | $x^{2} y + x^{2} t - x z t$ | |
| $=$ | $x y z + x z t - z^{2} t$ | |
| $=$ | $x y t + x t^{2} - z t^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y + x^{3} z^{2} + 20 x^{2} y z^{2} + 125 x y^{2} z^{2} + 25 y z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ 10x^{6} - 13x^{4} + 250x^{2} + 156 $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:0:1)$, $(1:0:1:0:0)$, $(0:0:1:0:0)$, $(0:0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2197265625xz^{10}-703505859375xz^{8}t^{2}-12574670968750xz^{6}t^{4}-11679981113875xz^{4}t^{6}+2878975783380xz^{2}t^{8}+88848484174xt^{10}-174804687500yz^{9}t-6245890937500yz^{7}t^{3}-19692791585000yz^{5}t^{5}+2358928481145yz^{3}t^{7}+429889129810yzt^{9}+27539062500yw^{9}t-13040546875yw^{7}t^{3}+437183065000yw^{5}t^{5}+101566304370yw^{3}t^{7}-253274919700ywt^{9}+3906250000z^{11}+151611328125z^{10}w+690859375000z^{9}t^{2}+11998330078125z^{8}wt^{2}+12700831125000z^{7}t^{4}+89546197031250z^{6}wt^{4}+13301650792000z^{5}t^{6}+7615034970000z^{4}wt^{6}-2402517474480z^{3}t^{8}-5767149293285z^{2}wt^{8}+5126953125zw^{10}-1357519531250zw^{8}t^{2}+1221358125000zw^{6}t^{4}-705508766875zw^{4}t^{6}-75255047475zw^{2}t^{8}-303298969946zt^{10}-976562500w^{11}+29277343750w^{9}t^{2}-996444578125w^{7}t^{4}+161678973250w^{5}t^{6}-34513808550w^{3}t^{8}-214325485772wt^{10}}{t(812500000xz^{8}t+235500000xz^{6}t^{3}-64220000xz^{4}t^{5}-384240xz^{2}t^{7}+48185xt^{9}+156250000yz^{9}+1423750000yz^{7}t^{2}-297360000yz^{5}t^{4}-2290775yz^{3}t^{6}+263578yzt^{8}-39062500yw^{9}-3203125yw^{7}t^{2}+18890000yw^{5}t^{4}-3305175yw^{3}t^{6}+236428ywt^{8}-812500000z^{9}t-7718750000z^{8}wt-366750000z^{7}t^{3}+2264500000z^{6}wt^{3}+118120000z^{5}t^{5}-38078000z^{4}wt^{5}-3763760z^{3}t^{7}+86385z^{2}wt^{7}-248046875zw^{8}t-38468750zw^{6}t^{3}+23057000zw^{4}t^{5}+701140zw^{2}t^{7}+125788zt^{9}-41015625w^{9}t+26062500w^{7}t^{3}+3447500w^{5}t^{5}-818475w^{3}t^{7}+173973wt^{9})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 20.72.3.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{25}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y+X^{3}Z^{2}+20X^{2}YZ^{2}+125XY^{2}Z^{2}+25YZ^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 20.72.3.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y^{4}-10y^{2}w^{2}-5yw^{2}t-13w^{4}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
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.b.1.1 | $40$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 40.72.1-10.a.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 40.72.1-10.a.1.2 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.288.5-20.f.1.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-20.f.2.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-20.h.1.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-20.h.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.q.1.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.q.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.w.1.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.5-40.w.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 40.288.7-20.d.1.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 40.288.7-20.d.1.3 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 40.288.7-20.e.1.1 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 40.288.7-20.e.1.5 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 40.288.7-40.g.1.1 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 40.288.7-40.g.1.9 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 40.288.7-40.j.1.1 | $40$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
| 40.288.7-40.j.1.9 | $40$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
| 40.288.7-20.m.1.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 40.288.7-20.m.1.5 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 40.288.7-20.m.2.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 40.288.7-20.m.2.2 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 40.288.7-40.bh.1.5 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 40.288.7-40.bh.1.7 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 40.288.7-40.bh.2.5 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 40.288.7-40.bh.2.6 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 40.720.19-20.f.1.3 | $40$ | $5$ | $5$ | $19$ | $1$ | $1^{16}$ |
| 120.288.5-60.bp.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bp.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.br.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.br.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eu.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eu.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fa.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fa.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.7-60.bc.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.bc.1.16 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.be.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.be.1.16 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ep.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ep.1.24 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.es.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.es.1.9 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.es.2.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.es.2.5 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ev.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ev.1.24 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdk.1.5 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdk.1.13 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdk.2.9 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdk.2.11 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.432.15-60.f.1.24 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 280.288.5-140.n.1.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.n.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.p.1.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.p.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bo.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bo.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bu.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bu.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.7-140.k.1.5 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.k.1.13 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.l.1.5 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.l.1.13 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.r.1.5 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.r.1.13 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.r.2.9 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.r.2.11 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.w.1.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.w.1.17 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.z.1.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.z.1.17 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bx.1.17 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bx.1.21 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bx.2.17 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bx.2.19 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |