Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20G3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.3.104 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&37\\30&39\end{bmatrix}$, $\begin{bmatrix}9&12\\32&19\end{bmatrix}$, $\begin{bmatrix}25&19\\6&33\end{bmatrix}$, $\begin{bmatrix}39&2\\10&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.72.3.ek.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $32$ |
| Full 40-torsion field degree: | $5120$ |
Jacobian
| Conductor: | $2^{18}\cdot5^{4}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 320.2.c.c, 1600.2.a.o |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} t - x z t - y w t $ |
| $=$ | $x z t - z^{2} t - z w t + w^{2} t$ | |
| $=$ | $x^{2} w - x z w - y w^{2}$ | |
| $=$ | $x z t - x w t - y z t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} - 3 x^{5} z - 7 x^{4} z^{2} - 10 x^{3} y^{2} z + 6 x^{3} z^{3} + 7 x^{2} z^{4} - 3 x z^{5} - z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -10x^{7} - 30x^{6} + 70x^{5} + 60x^{4} - 70x^{3} - 30x^{2} + 10x $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:0:1)$, $(0:1:0:0:0)$, $(1:2:-1:1:0)$, $(1:0:1:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3\cdot5^3}\cdot\frac{5859000000xw^{10}+641570000xw^{8}t^{2}-32689907000xw^{6}t^{4}-1582637400xw^{4}t^{6}+1910760xw^{2}t^{8}+12365xt^{10}+7000y^{7}t^{4}-14000y^{5}t^{6}+3850y^{3}t^{8}+10237500000y^{2}w^{9}+1966310000y^{2}w^{7}t^{2}-94118782000y^{2}w^{5}t^{4}-459727400y^{2}w^{3}t^{6}-279400y^{2}wt^{8}-23641800000yw^{10}-4147540000yw^{8}t^{2}+221020395000yw^{6}t^{4}-2040171900yw^{4}t^{6}-7790880yw^{2}t^{8}-1141yt^{10}-3254300000z^{2}w^{9}-493240000z^{2}w^{7}t^{2}+32713258000z^{2}w^{5}t^{4}+334438900z^{2}w^{3}t^{6}+371480z^{2}wt^{8}-3166800000zw^{10}-913240000zw^{8}t^{2}+32635972000zw^{6}t^{4}+4099373300zw^{4}t^{6}+5591770zw^{2}t^{8}-11217zt^{10}+3429300000w^{11}+351560000w^{9}t^{2}-32720523000w^{7}t^{4}+319888700w^{5}t^{6}-3744320w^{3}t^{8}-12729wt^{10}}{t^{2}w^{5}(640xw^{3}+38xwt^{2}+2560y^{2}w^{2}+7y^{2}t^{2}-5760yw^{3}+70ywt^{2}-640z^{2}w^{2}-7z^{2}t^{2}-640zw^{3}-108zwt^{2}+640w^{4}-3w^{2}t^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.72.3.ek.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{10}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}-3X^{5}Z-10X^{3}Y^{2}Z-7X^{4}Z^{2}+6X^{3}Z^{3}+7X^{2}Z^{4}-3XZ^{5}-Z^{6} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 40.72.3.ek.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z^{2}wt$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -w$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.72.0-10.a.2.4 | $10$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.72.0-10.a.2.13 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
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.m.1.4 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.288.5-40.bg.1.7 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.288.5-40.dg.1.7 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.dj.1.7 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.ek.1.11 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.eo.1.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.fz.1.11 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.288.5-40.gb.1.7 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.720.19-40.sa.1.3 | $40$ | $5$ | $5$ | $19$ | $4$ | $1^{8}\cdot2^{4}$ |
| 120.288.5-120.dbm.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dbp.1.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dco.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dcr.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ddq.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ddt.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.des.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dev.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.15-120.cgo.2.52 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 280.288.5-280.bvw.1.10 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bvy.1.13 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bwd.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bwf.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bya.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.byc.1.9 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.byh.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.byj.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |