Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $120$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $4 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $5^{2}\cdot10\cdot40$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4,-16$) |
Other labels
| Cummins and Pauli (CP) label: | 40B4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.120.4.69 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}27&0\\12&9\end{bmatrix}$, $\begin{bmatrix}29&16\\0&27\end{bmatrix}$, $\begin{bmatrix}29&21\\8&3\end{bmatrix}$, $\begin{bmatrix}33&12\\16&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.60.4.br.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{14}\cdot5^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 50.2.a.b$^{2}$, 1600.2.a.a, 1600.2.a.q |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 2 x^{2} - 2 x y + 4 y^{2} - z w $ |
| $=$ | $2 x^{3} + 4 x^{2} y + 2 x y^{2} + 2 x z^{2} - 2 x z w + 2 y z^{2} - y w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{6} - 5 x^{4} y^{2} + 9 x^{4} y z + 2 x^{4} z^{2} - 4 x^{2} y^{4} + 10 x^{2} y^{3} z + \cdots + 4 y z^{5} $ |
Rational points
This modular curve has 2 rational cusps and 2 rational CM points, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:1)$, $(0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{77578240xyz^{8}-331907072xyz^{7}w+630466560xyz^{6}w^{2}-672806912xyz^{5}w^{3}+400046400xyz^{4}w^{4}-95203904xyz^{3}w^{5}-24874392xyz^{2}w^{6}+20910952xyzw^{7}-4153154xyw^{8}-158236672y^{2}z^{8}+604704768y^{2}z^{7}w-887842816y^{2}z^{6}w^{2}+496334336y^{2}z^{5}w^{3}+196876800y^{2}z^{4}w^{4}-478608816y^{2}z^{3}w^{5}+309988384y^{2}z^{2}w^{6}-95793868y^{2}zw^{7}+12501708y^{2}w^{8}+262144z^{10}+28821504z^{9}w-128749568z^{8}w^{2}+231431424z^{7}w^{3}-192981440z^{6}w^{4}+23598216z^{5}w^{5}+102609204z^{4}w^{6}-100652978z^{3}w^{7}+45109343z^{2}w^{8}-10485760zw^{9}+1048576w^{10}}{8192xyz^{8}-20480xyz^{7}w-14336xyz^{6}w^{2}+29696xyz^{5}w^{3}-3200xyz^{4}w^{4}-10656xyz^{3}w^{5}+5216xyz^{2}w^{6}-732xyzw^{7}-2xyw^{8}+8192y^{2}z^{8}+28672y^{2}z^{7}w-36864y^{2}z^{6}w^{2}-9216y^{2}z^{5}w^{3}+25600y^{2}z^{4}w^{4}-5920y^{2}z^{3}w^{5}-3560y^{2}z^{2}w^{6}+1740y^{2}zw^{7}-180y^{2}w^{8}-5120z^{8}w^{2}+4608z^{7}w^{3}+1920z^{6}w^{4}-4144z^{5}w^{5}+1828z^{4}w^{6}-276z^{3}w^{7}-z^{2}w^{8}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.60.4.br.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ -2X^{6}-5X^{4}Y^{2}+9X^{4}YZ+2X^{4}Z^{2}-4X^{2}Y^{4}+10X^{2}Y^{3}Z-6X^{2}Y^{2}Z^{2}-4X^{2}Z^{4}+4Y^{3}Z^{3}-8Y^{2}Z^{4}+4YZ^{5} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 8.24.0-8.p.1.7 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-8.p.1.7 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.60.2-20.c.1.10 | $40$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 40.60.2-20.c.1.11 | $40$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.240.8-40.u.1.19 | $40$ | $2$ | $2$ | $8$ | $2$ | $1^{4}$ |
| 40.240.8-40.z.1.10 | $40$ | $2$ | $2$ | $8$ | $2$ | $1^{4}$ |
| 40.240.8-40.cf.1.5 | $40$ | $2$ | $2$ | $8$ | $5$ | $1^{4}$ |
| 40.240.8-40.cg.1.8 | $40$ | $2$ | $2$ | $8$ | $5$ | $1^{4}$ |
| 40.240.8-40.ct.1.5 | $40$ | $2$ | $2$ | $8$ | $4$ | $1^{4}$ |
| 40.240.8-40.cv.1.5 | $40$ | $2$ | $2$ | $8$ | $4$ | $1^{4}$ |
| 40.240.8-40.cx.1.3 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.240.8-40.cz.1.7 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.360.10-40.cx.1.25 | $40$ | $3$ | $3$ | $10$ | $5$ | $1^{6}$ |
| 40.480.13-40.oz.1.13 | $40$ | $4$ | $4$ | $13$ | $6$ | $1^{9}$ |
| 120.240.8-120.dr.1.12 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.dt.1.4 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ed.1.8 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ef.1.8 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fv.1.3 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fx.1.3 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gd.1.11 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gf.1.11 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.360.14-120.gh.1.33 | $120$ | $3$ | $3$ | $14$ | $?$ | not computed |
| 120.480.17-120.brp.1.53 | $120$ | $4$ | $4$ | $17$ | $?$ | not computed |
| 280.240.8-280.el.1.9 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.en.1.10 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.et.1.5 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.ev.1.14 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.fr.1.3 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.ft.1.3 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.fz.1.11 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.gb.1.11 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |