Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $400$ | ||
| 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: | $1$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-16$) |
Other labels
| Cummins and Pauli (CP) label: | 40B4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.120.4.164 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&27\\16&31\end{bmatrix}$, $\begin{bmatrix}19&14\\28&21\end{bmatrix}$, $\begin{bmatrix}21&39\\8&29\end{bmatrix}$, $\begin{bmatrix}37&34\\12&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.60.4.bq.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{10}\cdot5^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 50.2.a.b$^{2}$, 400.2.a.a, 400.2.a.f |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ x^{2} + x z + y w + 2 z^{2} $ |
| $=$ | $2 x^{2} z - x y w + 2 x w^{2} + y^{2} z - y z w - 2 z^{3} - 2 z w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{6} + 2 x^{4} y^{2} + 9 x^{4} y z - 5 x^{4} z^{2} + 2 x^{2} y^{4} + 3 x^{2} y^{2} z^{2} + \cdots + y^{3} z^{3} $ |
Rational points
This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:1:0:0)$, $(0:0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{247959xyz^{7}w+1069305xyz^{5}w^{3}+285486xyz^{3}w^{5}-106620xyzw^{7}-362824xz^{9}+2275690xz^{7}w^{2}+1375404xz^{5}w^{4}+604018xz^{3}w^{6}+223160xzw^{8}+128y^{10}-1280y^{9}w+5760y^{8}w^{2}-12800y^{7}w^{3}+8960y^{6}w^{4}+23040y^{5}w^{5}-75520y^{4}w^{6}+102400y^{3}w^{7}-67200y^{2}w^{8}+1499362yz^{8}w+1140415yz^{6}w^{3}-19449yz^{4}w^{5}-23530yz^{2}w^{7}+17540yw^{9}+377232z^{10}+1996906z^{8}w^{2}+2345416z^{6}w^{4}+633222z^{4}w^{6}-152680z^{2}w^{8}+32w^{10}}{8xyz^{7}w+15xyz^{5}w^{3}+4xyz^{3}w^{5}-xyzw^{7}+32xz^{9}+40xz^{7}w^{2}+18xz^{5}w^{4}-2xz^{3}w^{6}-6xzw^{8}-16yz^{8}w+10yz^{6}w^{3}+25yz^{4}w^{5}+10yz^{2}w^{7}-yw^{9}+32z^{8}w^{2}+42z^{6}w^{4}+26z^{4}w^{6}+2z^{2}w^{8}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.60.4.bq.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x-z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{6}+2X^{4}Y^{2}+9X^{4}YZ-5X^{4}Z^{2}+2X^{2}Y^{4}+3X^{2}Y^{2}Z^{2}-5X^{2}YZ^{3}+2X^{2}Z^{4}+Y^{5}Z-2Y^{4}Z^{2}+Y^{3}Z^{3} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.24.0-8.o.1.8 | $40$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 20.60.2-20.c.1.1 | $20$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 40.60.2-20.c.1.5 | $40$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
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.v.1.18 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-40.w.1.2 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-40.ce.1.2 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.240.8-40.cg.1.2 | $40$ | $2$ | $2$ | $8$ | $5$ | $1^{4}$ |
| 40.240.8-40.cs.1.1 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.240.8-40.cu.1.1 | $40$ | $2$ | $2$ | $8$ | $5$ | $1^{4}$ |
| 40.240.8-40.cw.1.1 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 40.240.8-40.cy.1.1 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 40.360.10-40.cw.1.11 | $40$ | $3$ | $3$ | $10$ | $2$ | $1^{6}$ |
| 40.480.13-40.oy.1.13 | $40$ | $4$ | $4$ | $13$ | $3$ | $1^{9}$ |
| 80.240.8-80.w.1.16 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.x.1.14 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.ba.1.15 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.8-80.bb.1.13 | $80$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 80.240.9-80.c.1.4 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.240.9-80.d.1.2 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.240.9-80.g.1.3 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.240.9-80.h.1.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.240.8-120.dq.1.15 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ds.1.11 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ec.1.9 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ee.1.11 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fu.1.12 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.fw.1.10 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gc.1.12 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ge.1.10 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.360.14-120.gg.1.50 | $120$ | $3$ | $3$ | $14$ | $?$ | not computed |
| 120.480.17-120.bro.1.19 | $120$ | $4$ | $4$ | $17$ | $?$ | not computed |
| 240.240.8-240.w.1.14 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.x.1.10 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.ba.1.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.8-240.bb.1.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.240.9-240.c.1.16 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.240.9-240.d.1.30 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.240.9-240.g.1.16 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.240.9-240.h.1.14 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.240.8-280.ek.1.16 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.em.1.12 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.es.1.10 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.eu.1.12 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.fq.1.12 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.fs.1.10 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.fy.1.12 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 280.240.8-280.ga.1.10 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |