Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $20$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $2^{6}\cdot10^{6}$ | Cusp orbits | $2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.1.496 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}13&18\\30&1\end{bmatrix}$, $\begin{bmatrix}13&38\\4&37\end{bmatrix}$, $\begin{bmatrix}19&18\\22&5\end{bmatrix}$, $\begin{bmatrix}25&22\\6&31\end{bmatrix}$, $\begin{bmatrix}27&34\\8&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.72.1.b.2 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: | $2^{2}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 20.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 10 x^{2} + 3 z^{2} + 4 z w $ |
| $=$ | $10 x y - 10 y^{2} + 2 z^{2} + 3 z w + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} - 12 x^{2} z^{2} + 8 x y z^{2} - 2 y^{2} z^{2} + 4 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(269z^{6}+172z^{5}w-1760z^{4}w^{2}-3840z^{3}w^{3}-3040z^{2}w^{4}-848zw^{5}+16w^{6})^{3}}{z(z+w)^{2}(2z+w)^{10}(3z+4w)^{5}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.72.1.b.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 5X^{4}-12X^{2}Z^{2}+8XYZ^{2}-2Y^{2}Z^{2}+4Z^{4} $ |
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.1.4 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.1-10.a.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
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.h.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.h.2.9 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.i.2.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.i.2.3 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.i.2.7 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.k.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.k.2.16 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.k.2.21 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.l.2.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.l.2.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.l.2.7 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.t.2.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.t.2.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.t.2.5 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.u.2.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.288.5-40.u.2.4 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.288.5-40.u.2.7 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.288.5-40.w.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.w.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.w.2.5 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.x.2.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.x.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.x.2.7 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.720.13-40.b.1.4 | $40$ | $5$ | $5$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
| 80.288.3-80.a.1.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.a.1.8 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.a.1.10 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.a.1.16 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.a.1.23 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.a.1.31 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.b.1.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.b.1.8 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.b.1.10 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.b.1.16 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.b.1.23 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.b.1.31 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.c.2.4 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.c.2.7 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.c.2.12 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.c.2.15 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.c.2.22 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.c.2.30 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.d.2.4 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.d.2.6 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.d.2.12 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.d.2.14 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.d.2.23 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.d.2.31 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.7-80.a.1.3 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.a.1.4 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.b.1.5 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.b.1.6 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.c.1.7 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.c.1.8 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.d.1.7 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.d.1.8 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.5-120.db.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.db.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.db.2.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dc.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dc.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dc.2.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.de.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.de.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.de.2.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.df.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.df.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.df.2.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.el.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.el.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.el.2.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.em.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.em.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.em.2.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eo.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eo.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eo.2.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ep.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ep.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ep.2.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.13-120.b.1.3 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 240.288.3-240.a.1.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.a.1.14 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.a.1.20 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.a.1.30 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.a.1.45 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.a.1.61 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.b.1.3 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.b.1.14 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.b.1.19 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.b.1.30 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.b.1.46 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.b.1.62 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.c.2.7 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.c.2.10 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.c.2.23 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.c.2.26 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.c.2.44 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.c.2.60 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.d.2.8 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.d.2.10 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.d.2.24 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.d.2.26 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.d.2.43 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.d.2.59 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.7-240.a.1.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.a.1.7 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.b.1.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.b.1.7 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.c.1.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.c.1.7 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.d.1.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.d.1.7 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.5-280.h.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.h.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.h.2.9 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.i.2.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.i.2.8 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.i.2.13 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.k.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.k.2.8 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.k.2.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.l.2.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.l.2.8 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.l.2.16 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.t.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.t.2.8 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.t.2.9 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.u.2.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.u.2.8 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.u.2.13 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.w.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.w.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.w.2.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.x.2.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.x.2.8 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.x.2.16 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |