Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $40$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $8$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8\cdot10^{2}\cdot20^{3}\cdot40$ | Cusp orbits | $1^{8}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40X7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.288.7.902 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&32\\4&35\end{bmatrix}$, $\begin{bmatrix}9&12\\16&35\end{bmatrix}$, $\begin{bmatrix}9&28\\38&19\end{bmatrix}$, $\begin{bmatrix}17&0\\4&33\end{bmatrix}$, $\begin{bmatrix}25&32\\24&23\end{bmatrix}$, $\begin{bmatrix}31&32\\0&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.144.7.v.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $2$ |
| Cyclic 40-torsion field degree: | $32$ |
| Full 40-torsion field degree: | $2560$ |
Jacobian
| Conductor: | $2^{19}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot4$ |
| Newforms: | 20.2.a.a$^{2}$, 40.2.a.a, 40.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ y u + z v $ |
| $=$ | $ - x v + z u$ | |
| $=$ | $x y + z^{2}$ | |
| $=$ | $w u + w v + t v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{5} y^{3} + 2 x^{5} y z^{2} + x^{4} y^{4} + 4 x^{4} y^{2} z^{2} - 4 x^{4} z^{4} - 2 x^{3} y^{5} + \cdots + y^{8} $ |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(1/2:-1/2:-1/2:1:0:-1:1)$, $(-1/2:1/2:1/2:1:0:-1:1)$, $(0:0:0:0:1:0:0)$, $(0:0:0:0:0:0:1)$, $(0:0:0:0:0:1:1)$, $(0:0:0:1:0:0:0)$, $(1/2:-1/2:1/2:1:-2:1:1)$, $(-1/2:1/2:-1/2:1:-2:1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{10066907092648z^{2}v^{10}+32w^{12}-512w^{11}v+27200w^{10}v^{2}-391936w^{9}v^{3}+8882400w^{8}v^{4}-109895680w^{7}v^{5}+1396950912w^{6}v^{6}-13678644736w^{5}v^{7}+111183813088w^{4}v^{8}-773800144384w^{3}v^{9}+992115w^{2}t^{10}-96104362w^{2}t^{9}v+953743994w^{2}t^{8}v^{2}-1312539480w^{2}t^{7}v^{3}+1153483416w^{2}t^{6}v^{4}-15167715448w^{2}t^{5}v^{5}+13368956510w^{2}t^{4}v^{6}-53427211442w^{2}t^{3}v^{7}+209060949226w^{2}t^{2}v^{8}-592124107022w^{2}tv^{9}+1102063823372w^{2}v^{10}+987363wt^{11}-110019016wt^{10}v+1301417508wt^{9}v^{2}-2960890460wt^{8}v^{3}+4225980524wt^{7}v^{4}-22533294880wt^{6}v^{5}+29408433270wt^{5}v^{6}-137574129238wt^{4}v^{7}+354932363388wt^{3}v^{8}-18522684102wt^{2}v^{9}+6068278592427wtv^{10}+3612717598816wv^{11}+500000t^{12}+7012637t^{11}u-56007885t^{11}v-186014143t^{10}uv+734436682t^{10}v^{2}+1029462134t^{9}uv^{2}-2276660894t^{9}v^{3}-2068619098t^{8}uv^{3}+3762542536t^{8}v^{4}+4566651472t^{7}uv^{4}-13456850364t^{7}v^{5}-13550800700t^{6}uv^{5}+27549371459t^{6}v^{6}+46633569058t^{5}uv^{6}-97566566569t^{5}v^{7}-84061732065t^{4}uv^{7}+241116043456t^{4}v^{8}+132833109542t^{3}uv^{8}-98674893695t^{3}v^{9}-1448321023757t^{2}uv^{9}+4039182839626t^{2}v^{10}+453246302947tuv^{10}+2810967742609tv^{11}+6513149u^{12}+59850518u^{11}v-642411824u^{10}v^{2}+1417092370u^{9}v^{3}+7447861557u^{8}v^{4}-45170397492u^{7}v^{5}+69517963775u^{6}v^{6}+206776604906u^{5}v^{7}-1017397956557u^{4}v^{8}+2313417639230u^{3}v^{9}-2337182115999u^{2}v^{10}+801749856335uv^{11}+32v^{12}}{43349592z^{2}v^{10}-512w^{10}v^{2}+8192w^{9}v^{3}-55296w^{8}v^{4}+196608w^{7}v^{5}-352256w^{6}v^{6}+69632w^{5}v^{7}+1382400w^{4}v^{8}-5492736w^{3}v^{9}+73w^{2}t^{10}+966w^{2}t^{9}v+3486w^{2}t^{8}v^{2}+88w^{2}t^{7}v^{3}-6008w^{2}t^{6}v^{4}-52232w^{2}t^{5}v^{5}+15730w^{2}t^{4}v^{6}+305314w^{2}t^{3}v^{7}-53690w^{2}t^{2}v^{8}-7168562w^{2}tv^{9}+3650420w^{2}v^{10}+117wt^{11}+176wt^{10}v+2460wt^{9}v^{2}-1404wt^{8}v^{3}-9516wt^{7}v^{4}+18176wt^{6}v^{5}-40774wt^{5}v^{6}+14102wt^{4}v^{7}-19164wt^{3}v^{8}-1450538wt^{2}v^{9}+32728965wtv^{10}+23862272wv^{11}-117t^{11}u+73t^{11}v+131t^{10}uv+586t^{10}v^{2}-1346t^{9}uv^{2}+2606t^{9}v^{3}+6002t^{8}uv^{3}-6296t^{8}v^{4}-5328t^{7}uv^{4}+9516t^{7}v^{5}-12692t^{6}uv^{5}-56875t^{6}v^{6}-32106t^{5}uv^{6}+20857t^{5}v^{7}+203833t^{4}uv^{7}+51280t^{4}v^{8}-56126t^{3}uv^{8}-287385t^{3}v^{9}-9749035t^{2}uv^{9}+23197742t^{2}v^{10}-1995555tuv^{10}+19501935tv^{11}-117u^{12}+1346u^{11}v-7056u^{10}v^{2}+12550u^{9}v^{3}+50179u^{8}v^{4}-311036u^{7}v^{5}+394881u^{6}v^{6}+1444998u^{5}v^{7}-5865371u^{4}v^{8}+11544866u^{3}v^{9}-11625577u^{2}v^{10}+4360337uv^{11}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.144.7.v.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ -X^{5}Y^{3}+2X^{5}YZ^{2}+X^{4}Y^{4}+4X^{4}Y^{2}Z^{2}-4X^{4}Z^{4}-2X^{3}Y^{5}+4X^{3}Y^{3}Z^{2}-4X^{3}YZ^{4}+2X^{2}Y^{6}-4X^{2}Y^{4}Z^{2}-XY^{7}+2XY^{5}Z^{2}+Y^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.144.3-20.b.1.7 | $20$ | $2$ | $2$ | $3$ | $0$ | $4$ |
| 40.48.0-8.e.1.9 | $40$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 40.144.3-20.b.1.25 | $40$ | $2$ | $2$ | $3$ | $0$ | $4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.576.13-40.db.1.17 | $40$ | $2$ | $2$ | $13$ | $0$ | $2\cdot4$ |
| 40.576.13-40.db.2.17 | $40$ | $2$ | $2$ | $13$ | $0$ | $2\cdot4$ |
| 40.576.13-40.df.2.6 | $40$ | $2$ | $2$ | $13$ | $0$ | $2\cdot4$ |
| 40.576.13-40.df.4.7 | $40$ | $2$ | $2$ | $13$ | $0$ | $2\cdot4$ |
| 40.576.13-40.dl.3.2 | $40$ | $2$ | $2$ | $13$ | $0$ | $2\cdot4$ |
| 40.576.13-40.dl.4.3 | $40$ | $2$ | $2$ | $13$ | $0$ | $2\cdot4$ |
| 40.576.13-40.dp.3.5 | $40$ | $2$ | $2$ | $13$ | $0$ | $2\cdot4$ |
| 40.576.13-40.dp.4.5 | $40$ | $2$ | $2$ | $13$ | $0$ | $2\cdot4$ |
| 40.576.15-40.f.1.10 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{4}\cdot4$ |
| 40.576.15-40.i.1.5 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{4}\cdot4$ |
| 40.576.15-40.o.1.8 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{4}\cdot4$ |
| 40.576.15-40.r.1.9 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{4}\cdot4$ |
| 40.576.15-40.ba.2.3 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{4}\cdot4$ |
| 40.576.15-40.bc.2.9 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{4}\cdot4$ |
| 40.576.15-40.bh.1.16 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{4}\cdot4$ |
| 40.576.15-40.bj.2.1 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{4}\cdot4$ |
| 40.576.15-40.bp.2.11 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.bp.3.6 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.br.3.13 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.br.4.10 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.bx.3.11 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.bx.4.6 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.bz.2.11 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.bz.3.6 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.cj.2.9 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.cj.4.2 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.cl.2.10 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.cl.3.4 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.cr.2.9 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.cr.3.2 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.ct.2.10 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.ct.4.4 | $40$ | $2$ | $2$ | $15$ | $0$ | $2^{2}\cdot4$ |
| 40.576.15-40.cy.2.11 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{4}\cdot4$ |
| 40.576.15-40.da.2.6 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{4}\cdot4$ |
| 40.576.15-40.df.2.3 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{4}\cdot4$ |
| 40.576.15-40.dh.2.14 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{4}\cdot4$ |
| 40.576.15-40.dn.2.5 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{4}\cdot4$ |
| 40.576.15-40.dq.1.12 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{4}\cdot4$ |
| 40.576.15-40.dw.1.10 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{4}\cdot4$ |
| 40.576.15-40.dz.2.10 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{4}\cdot4$ |
| 40.576.17-40.cc.2.7 | $40$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.ci.1.7 | $40$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.cu.1.8 | $40$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.cx.1.5 | $40$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.eu.1.7 | $40$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.ew.1.7 | $40$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.fd.2.16 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.ff.1.5 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2\cdot4$ |
| 40.576.17-40.gl.2.7 | $40$ | $2$ | $2$ | $17$ | $0$ | $2\cdot4^{2}$ |
| 40.576.17-40.gl.3.13 | $40$ | $2$ | $2$ | $17$ | $0$ | $2\cdot4^{2}$ |
| 40.576.17-40.gn.2.1 | $40$ | $2$ | $2$ | $17$ | $0$ | $2\cdot4^{2}$ |
| 40.576.17-40.gn.4.1 | $40$ | $2$ | $2$ | $17$ | $0$ | $2\cdot4^{2}$ |
| 40.576.17-40.gt.2.5 | $40$ | $2$ | $2$ | $17$ | $0$ | $2\cdot4^{2}$ |
| 40.576.17-40.gt.4.9 | $40$ | $2$ | $2$ | $17$ | $0$ | $2\cdot4^{2}$ |
| 40.576.17-40.gv.2.5 | $40$ | $2$ | $2$ | $17$ | $0$ | $2\cdot4^{2}$ |
| 40.576.17-40.gv.3.9 | $40$ | $2$ | $2$ | $17$ | $0$ | $2\cdot4^{2}$ |
| 40.1440.43-40.et.2.3 | $40$ | $5$ | $5$ | $43$ | $1$ | $1^{16}\cdot2^{4}\cdot4^{3}$ |