Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $80$ | ||
| 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 | $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
| 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: | 20H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.1.664 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&4\\12&17\end{bmatrix}$, $\begin{bmatrix}11&12\\34&9\end{bmatrix}$, $\begin{bmatrix}23&9\\26&11\end{bmatrix}$, $\begin{bmatrix}33&36\\6&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.72.1.p.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: | $2^{4}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 80.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 10 x^{2} + z^{2} + z w $ |
| $=$ | $10 y^{2} - 4 z^{2} - 8 z w - 5 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 4 x^{2} z^{2} - 2 y^{2} z^{2} + 20 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(16z^{6}+80z^{5}w+160z^{4}w^{2}+160z^{3}w^{3}+80z^{2}w^{4}+20zw^{5}+5w^{6})^{3}}{w^{5}z^{2}(z+w)^{10}(4z+5w)}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.72.1.p.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{10}z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+4X^{2}Z^{2}-2Y^{2}Z^{2}+20Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.72.1-20.b.1.5 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.1-20.b.1.9 | $40$ | $2$ | $2$ | $1$ | $0$ | 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.1.7 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.bc.1.3 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.288.5-40.cl.1.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.cp.1.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.ej.2.3 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.em.2.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.fd.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.ff.2.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.720.13-40.bb.1.4 | $40$ | $5$ | $5$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 80.288.3-80.m.2.7 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.m.2.15 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.n.2.7 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.n.2.15 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.q.1.11 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.q.1.15 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.r.1.11 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.3-80.r.1.15 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.288.7-80.bg.1.13 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.bg.1.15 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.bh.1.13 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.bh.1.15 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.bk.2.13 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.bk.2.15 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.bl.2.13 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 80.288.7-80.bl.2.15 | $80$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.5-120.bch.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bcj.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bcv.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bcx.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.boh.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.boj.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bov.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.box.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.13-120.dv.1.15 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 240.288.3-240.m.2.26 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.m.2.30 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.n.2.22 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.n.2.30 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.q.1.22 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.q.1.30 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.r.1.26 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.3-240.r.1.30 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.7-240.tw.1.22 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.tw.1.30 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.tx.1.14 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.tx.1.30 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.ua.2.22 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.ua.2.30 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.ub.2.26 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-240.ub.2.30 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.5-280.pr.1.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.ps.1.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.py.1.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.pz.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.rv.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.rw.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.sc.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.sd.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |