Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $320$ | ||
| 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: | $1$ | ||||||
| $\Q$-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: | 40.144.3.406 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&6\\30&7\end{bmatrix}$, $\begin{bmatrix}7&6\\30&33\end{bmatrix}$, $\begin{bmatrix}31&16\\0&7\end{bmatrix}$, $\begin{bmatrix}33&18\\30&1\end{bmatrix}$, $\begin{bmatrix}37&16\\6&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.72.3.a.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: | $2^{14}\cdot5^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}$ |
| Newforms: | 20.2.a.a, 320.2.a.c, 320.2.a.f |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x z^{2} + y z^{2} + y z w $ |
| $=$ | $x z w + y z w + y w^{2}$ | |
| $=$ | $x y z + y^{2} z + y^{2} w$ | |
| $=$ | $x z t + y z t + y w t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} y - 4 x^{4} z + 8 x^{2} y z^{2} - 4 x^{2} z^{3} - 10 y^{2} z^{3} + 11 y z^{4} - 3 z^{5} $ |
Weierstrass model Weierstrass model
| $ y^{2} + x^{4} y $ | $=$ | $ 4x^{6} - 2x^{4} + 16x^{2} + 4 $ |
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 |
|---|
| $(0:1:0:0:0)$, $(0:0:1:-1:1)$, $(0:0:0:0:1)$, $(0:0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^3}\cdot\frac{512xw^{10}-25600xw^{9}t-526080xw^{8}t^{2}-2606140xw^{7}t^{3}-3274700xw^{6}t^{4}+2133118xw^{5}t^{5}+4458910xw^{4}t^{6}+1298370xw^{3}t^{7}-34490xw^{2}t^{8}+58500xwt^{9}+32xt^{10}+200y^{9}t^{2}-2300y^{7}t^{4}+11250y^{5}t^{6}-28475y^{3}t^{8}-17408yzw^{9}-498688yzw^{8}t-3469120yzw^{7}t^{2}-9039512yzw^{6}t^{3}-8508392yzw^{5}t^{4}-3872112yzw^{4}t^{5}-1857040yzw^{3}t^{6}-975468yzw^{2}t^{7}-117428yzwt^{8}-25600yzt^{9}-34304yw^{10}-670208yw^{9}t-3395912yw^{8}t^{2}-4267084yw^{7}t^{3}+5656708yw^{6}t^{4}+13342846yw^{5}t^{5}+8014562yw^{4}t^{6}+1658034yw^{3}t^{7}+230772yw^{2}t^{8}-22822ywt^{9}+11732yt^{10}}{w^{2}(500xw^{7}t+1901xw^{6}t^{2}+2696xw^{5}t^{3}+1818xw^{4}t^{4}+560xw^{3}t^{5}+28xw^{2}t^{6}-24xwt^{7}-4xt^{8}+300yzw^{7}+1400yzw^{6}t+2366yzw^{5}t^{2}+1962yzw^{4}t^{3}+864yzw^{3}t^{4}+192yzw^{2}t^{5}+16yzwt^{6}+600yw^{8}+2600yw^{7}t+4233yw^{6}t^{2}+3309yw^{5}t^{3}+1174yw^{4}t^{4}+10yw^{3}t^{5}-132yw^{2}t^{6}-40ywt^{7}-4yt^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.72.3.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2z$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{4}Y-4X^{4}Z+8X^{2}YZ^{2}-4X^{2}Z^{3}-10Y^{2}Z^{3}+11YZ^{4}-3Z^{5} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 40.72.3.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4y^{2}z^{2}+22z^{4}-20z^{3}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.72.1-10.a.1.2 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 40.24.0-8.a.1.2 | $40$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 40.72.1-10.a.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.288.5-40.a.1.5 | $40$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 40.288.5-40.a.2.6 | $40$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 40.288.5-40.c.1.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 40.288.5-40.c.2.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 40.288.5-40.g.1.3 | $40$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 40.288.5-40.g.2.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 40.288.5-40.i.1.3 | $40$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 40.288.5-40.i.2.3 | $40$ | $2$ | $2$ | $5$ | $1$ | $2$ |
| 40.288.7-40.a.1.5 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 40.288.7-40.a.1.6 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 40.288.7-40.c.1.6 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 40.288.7-40.c.1.10 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 40.288.7-40.e.1.6 | $40$ | $2$ | $2$ | $7$ | $4$ | $1^{4}$ |
| 40.288.7-40.e.1.13 | $40$ | $2$ | $2$ | $7$ | $4$ | $1^{4}$ |
| 40.288.7-40.g.1.2 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 40.288.7-40.g.1.6 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 40.288.7-40.x.1.2 | $40$ | $2$ | $2$ | $7$ | $1$ | $2^{2}$ |
| 40.288.7-40.x.1.3 | $40$ | $2$ | $2$ | $7$ | $1$ | $2^{2}$ |
| 40.288.7-40.x.2.2 | $40$ | $2$ | $2$ | $7$ | $1$ | $2^{2}$ |
| 40.288.7-40.x.2.3 | $40$ | $2$ | $2$ | $7$ | $1$ | $2^{2}$ |
| 40.288.7-40.z.1.2 | $40$ | $2$ | $2$ | $7$ | $1$ | $2^{2}$ |
| 40.288.7-40.z.1.3 | $40$ | $2$ | $2$ | $7$ | $1$ | $2^{2}$ |
| 40.288.7-40.z.2.2 | $40$ | $2$ | $2$ | $7$ | $1$ | $2^{2}$ |
| 40.288.7-40.z.2.3 | $40$ | $2$ | $2$ | $7$ | $1$ | $2^{2}$ |
| 40.720.19-40.a.1.2 | $40$ | $5$ | $5$ | $19$ | $6$ | $1^{16}$ |
| 120.288.5-120.dg.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dg.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.di.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.di.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dm.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dm.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.do.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.do.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.7-120.cq.1.9 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.cq.1.14 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.cs.1.10 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.cs.1.22 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ef.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ef.1.22 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.eh.1.18 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.eh.1.30 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bco.1.3 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bco.1.10 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bco.2.4 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bco.2.5 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcq.1.6 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcq.1.9 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcq.2.4 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcq.2.9 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.432.15-120.a.1.2 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 280.288.5-280.y.1.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.y.2.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.ba.1.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.ba.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.be.1.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.be.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bg.1.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bg.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.7-280.d.1.18 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.d.1.21 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.f.1.13 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.f.1.21 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.k.1.13 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.k.1.26 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.m.1.5 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.m.1.13 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bn.1.4 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bn.1.5 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bn.2.3 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bn.2.6 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bp.1.4 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bp.1.5 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bp.2.4 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bp.2.5 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |