Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $80$ | ||
| 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: | 20I3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.3.1026 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&30\\26&1\end{bmatrix}$, $\begin{bmatrix}11&10\\20&31\end{bmatrix}$, $\begin{bmatrix}35&22\\16&11\end{bmatrix}$, $\begin{bmatrix}35&24\\6&13\end{bmatrix}$, $\begin{bmatrix}39&8\\32&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.72.3.f.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: | $2^{10}\cdot5^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 20.2.a.a, 80.2.c.a |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x y w + x y t + y^{2} w + y z w + y z t $ |
| $=$ | $x^{2} w + x^{2} t + x y w - y z w - z^{2} w - z^{2} t$ | |
| $=$ | $x w t + x t^{2} + y w t + z w t + z t^{2}$ | |
| $=$ | $x w^{2} + x w t + y w^{2} + z w^{2} + z w t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} y + 5 x^{4} z - x^{2} y^{2} z + 2 x^{2} y z^{2} + 2 x^{2} z^{3} + y z^{4} + z^{5} $ |
Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ 2x^{6} + 5x^{4} + 10x^{2} + 6 $ |
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 |
|---|
| $(0:0:0:0:1)$, $(-1:0:1:0:0)$, $(0:1:0:0:0)$, $(0:0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{87890625xz^{10}+578125000xz^{8}t^{2}+86171875xz^{6}t^{4}+251775000xz^{4}t^{6}-33231750xz^{2}t^{8}-67996696xt^{10}+390625y^{9}t^{2}+46875y^{7}t^{4}+440625y^{5}t^{6}+52250y^{3}t^{8}+11260928yw^{10}-27849728yw^{9}t+164748551yw^{8}t^{2}+211796536yw^{7}t^{3}+433650117yw^{6}t^{4}+214047920yw^{5}t^{5}+231843771yw^{4}t^{6}-25482990yw^{3}t^{7}+74327535yw^{2}t^{8}-67373460ywt^{9}+162922yt^{10}-156250000z^{11}+267578125z^{9}t^{2}-166953125z^{7}t^{4}+205712500z^{5}t^{6}-2068000z^{3}t^{8}-60264448zw^{10}-218419200zw^{9}t-429891596zw^{8}t^{2}-609869164zw^{7}t^{3}-526101487zw^{6}t^{4}-349365227zw^{5}t^{5}-126236333zw^{4}t^{6}-43877811zw^{3}t^{7}+5902592zw^{2}t^{8}-6360298zwt^{9}-67833774zt^{10}}{250000xz^{6}t^{4}-100000xz^{4}t^{6}-800xz^{2}t^{8}+4064xt^{10}-625y^{5}t^{6}-75y^{3}t^{8}-7424yw^{10}+17024yw^{9}t+123776yw^{8}t^{2}+167904yw^{7}t^{3}+95264yw^{6}t^{4}+25368yw^{5}t^{5}-10635yw^{4}t^{6}+2758yw^{3}t^{7}-3137yw^{2}t^{8}+4293ywt^{9}+47yt^{10}+250000z^{7}t^{4}-150000z^{5}t^{6}-10800z^{3}t^{8}+35584zw^{10}+126400zw^{9}t+181824zw^{8}t^{2}+122896zw^{7}t^{3}+19840zw^{6}t^{4}-28124zw^{5}t^{5}-22526zw^{4}t^{6}-819zw^{3}t^{7}+2099zw^{2}t^{8}+2355zwt^{9}+4111zt^{10}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 20.72.3.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 5X^{4}Y+5X^{4}Z-X^{2}Y^{2}Z+2X^{2}YZ^{2}+2X^{2}Z^{3}+YZ^{4}+Z^{5} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 20.72.3.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x^{4}+x^{2}w^{2}-x^{2}wt$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.72.1-10.a.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 40.72.1-10.a.1.4 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
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.a.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.c.2.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-20.d.1.5 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-20.e.1.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-20.h.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.l.1.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.o.1.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.288.5-40.x.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.7-20.i.1.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-20.i.1.3 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-20.j.2.12 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-20.j.2.16 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-20.l.1.6 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-20.l.1.8 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-20.m.1.2 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-20.m.1.4 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.z.2.7 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.z.2.15 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.bc.2.3 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.bc.2.11 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.bg.2.8 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{2}\cdot2$ |
| 40.288.7-40.bg.2.16 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{2}\cdot2$ |
| 40.288.7-40.bj.2.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.bj.2.9 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.720.19-20.r.1.3 | $40$ | $5$ | $5$ | $19$ | $0$ | $1^{6}\cdot2^{5}$ |
| 120.288.5-60.bg.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bj.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bs.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bv.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.du.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ed.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fe.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fn.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.7-60.en.1.2 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.en.1.5 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.ep.1.12 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.ep.1.16 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.eu.2.6 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.eu.2.13 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.ew.1.12 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.ew.1.16 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcz.2.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcz.2.23 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdf.1.21 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdf.1.29 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bds.1.20 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bds.1.27 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdy.2.10 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdy.2.26 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.432.15-60.x.2.11 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 280.288.5-140.r.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.t.1.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.v.1.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.x.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cb.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.ch.1.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cn.1.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.ct.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.7-140.u.1.4 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.u.1.7 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.v.2.4 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.v.2.7 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.y.2.8 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.y.2.13 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.z.1.2 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.z.1.4 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cq.2.14 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cq.2.29 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.ct.2.4 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.ct.2.11 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.dc.2.16 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.dc.2.29 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.df.2.3 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.df.2.11 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |