Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $10^{4}\cdot40^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $5$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-16$) |
Other labels
| Cummins and Pauli (CP) label: | 40A8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.8.152 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&16\\6&9\end{bmatrix}$, $\begin{bmatrix}37&20\\5&27\end{bmatrix}$, $\begin{bmatrix}39&8\\20&33\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.8.cg.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{34}\cdot5^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{8}$ |
| Newforms: | 50.2.a.b$^{2}$, 400.2.a.a, 400.2.a.f, 1600.2.a.a$^{2}$, 1600.2.a.q$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x r - y v + 2 t u $ |
| $=$ | $x w - 2 y u - z v$ | |
| $=$ | $x w + x u + y u + z v - t v - t r$ | |
| $=$ | $2 x v + y v + y r + w t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 1024 x^{10} - 1152 x^{8} y^{2} + 576 x^{8} z^{2} - 260 x^{6} y^{4} + 676 x^{6} y^{2} z^{2} + \cdots + 2 y^{4} z^{6} $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 40.60.4.w.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle v$ |
| $\displaystyle W$ | $=$ | $\displaystyle -r$ |
Equation of the image curve:
| $0$ | $=$ | $ 14X^{2}+2Y^{2}+Z^{2}-W^{2} $ |
| $=$ | $ 2X^{3}-2XY^{2}-XZ^{2}+YZW $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.120.8.cg.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x-y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ -1024X^{10}-1152X^{8}Y^{2}+576X^{8}Z^{2}-260X^{6}Y^{4}+676X^{6}Y^{2}Z^{2}-209X^{6}Z^{4}-140X^{4}Y^{6}+98X^{4}Y^{4}Z^{2}-86X^{4}Y^{2}Z^{4}+36X^{4}Z^{6}-12X^{2}Y^{8}+8X^{2}Y^{6}Z^{2}-13X^{2}Y^{4}Z^{4}+12X^{2}Y^{2}Z^{6}-4X^{2}Z^{8}-4Y^{10}+10Y^{8}Z^{2}-8Y^{6}Z^{4}+2Y^{4}Z^{6} $ |
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 |
|---|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.y.1.3 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.y.1.3 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.120.4-40.w.1.2 | $40$ | $2$ | $2$ | $4$ | $2$ | $1^{4}$ |
| 40.120.4-40.w.1.8 | $40$ | $2$ | $2$ | $4$ | $2$ | $1^{4}$ |
| 40.120.4-40.bq.1.3 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
| 40.120.4-40.bq.1.8 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
| 40.120.4-40.br.1.13 | $40$ | $2$ | $2$ | $4$ | $2$ | $1^{4}$ |
| 40.120.4-40.br.1.16 | $40$ | $2$ | $2$ | $4$ | $2$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.720.22-40.fe.1.16 | $40$ | $3$ | $3$ | $22$ | $12$ | $1^{14}$ |
| 40.960.29-40.qu.1.8 | $40$ | $4$ | $4$ | $29$ | $15$ | $1^{21}$ |
| 80.480.16-80.bo.1.6 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.bp.1.6 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.bq.1.8 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.br.1.8 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.17-80.bc.1.1 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.bd.1.2 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.be.1.3 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.bf.1.4 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.18-80.r.1.16 | $80$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 80.480.18-80.z.1.16 | $80$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 80.480.18-80.bh.1.16 | $80$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 80.480.18-80.bk.1.16 | $80$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 240.480.16-240.bo.1.14 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.bp.1.12 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.bq.1.16 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.br.1.16 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.17-240.bc.1.3 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.bd.1.4 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.be.1.3 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.bf.1.4 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.18-240.t.1.31 | $240$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 240.480.18-240.z.1.23 | $240$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 240.480.18-240.bh.1.31 | $240$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 240.480.18-240.bk.1.31 | $240$ | $2$ | $2$ | $18$ | $?$ | not computed |