Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $400$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $7 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $10^{4}\cdot20^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20B7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.7.724 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&2\\36&9\end{bmatrix}$, $\begin{bmatrix}7&30\\20&7\end{bmatrix}$, $\begin{bmatrix}13&24\\36&17\end{bmatrix}$, $\begin{bmatrix}25&22\\22&25\end{bmatrix}$, $\begin{bmatrix}35&36\\14&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.120.7.f.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{20}\cdot5^{14}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 50.2.a.b$^{2}$, 100.2.a.a, 400.2.a.a, 400.2.a.b, 400.2.a.f, 400.2.a.g |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x t + x u - x v + z w $ |
| $=$ | $x w + x t - x u - y w + z w$ | |
| $=$ | $y t + y u - y v + z w - 2 z u + z v$ | |
| $=$ | $x z - 3 y z - z^{2} + t u + u^{2} - u v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 25 x^{12} + 350 x^{10} y^{2} + 1225 x^{8} y^{4} - 300 x^{8} y^{2} z^{2} - 10 x^{8} z^{4} + \cdots + y^{4} z^{8} $ |
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
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 10.60.3.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -2x+y-3z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4x-2y+z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x+2y-z$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}-3X^{3}Y-5X^{2}Y^{2}-4XY^{3}-2Y^{4}+3X^{3}Z-18X^{2}YZ-17XY^{2}Z+4Y^{3}Z-5X^{2}Z^{2}+17XYZ^{2}-6Y^{2}Z^{2}+4XZ^{3}+4YZ^{3}-2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.120.7.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y+z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 25X^{12}+350X^{10}Y^{2}+1225X^{8}Y^{4}-300X^{8}Y^{2}Z^{2}-10X^{8}Z^{4}-2600X^{6}Y^{4}Z^{2}-145X^{6}Y^{2}Z^{4}+550X^{4}Y^{4}Z^{4}+50X^{4}Y^{2}Z^{6}+X^{4}Z^{8}-40X^{2}Y^{4}Z^{6}-7X^{2}Y^{2}Z^{8}+Y^{4}Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.120.3-10.a.1.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.120.3-10.a.1.5 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.480.13-20.e.1.2 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 40.480.13-20.h.1.2 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.13-20.i.1.2 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.13-20.l.1.1 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 40.480.13-40.o.1.8 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-40.x.1.8 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-40.ba.1.4 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
| 40.480.13-40.bj.1.4 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.15-20.g.1.2 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-20.h.1.5 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-20.h.1.8 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-20.m.1.1 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-20.m.1.4 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-20.n.1.2 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-20.n.1.3 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.bd.1.3 | $40$ | $2$ | $2$ | $15$ | $7$ | $1^{8}$ |
| 40.480.15-40.bd.1.8 | $40$ | $2$ | $2$ | $15$ | $7$ | $1^{8}$ |
| 40.480.15-40.bg.1.3 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.bg.1.8 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.bs.1.2 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.bs.1.8 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.bv.1.2 | $40$ | $2$ | $2$ | $15$ | $8$ | $1^{8}$ |
| 40.480.15-40.bv.1.8 | $40$ | $2$ | $2$ | $15$ | $8$ | $1^{8}$ |
| 40.720.19-20.t.1.4 | $40$ | $3$ | $3$ | $19$ | $3$ | $1^{12}$ |
| 120.480.13-60.u.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.x.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.bg.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.bj.1.4 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.ck.1.10 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.ct.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.du.1.12 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.ed.1.12 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.15-60.o.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.o.1.11 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.q.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.q.1.15 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.bd.1.5 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.bd.1.15 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.be.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.be.1.11 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cc.1.15 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cc.1.19 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ci.1.15 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ci.1.19 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ds.1.15 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ds.1.21 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dv.1.15 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.dv.1.21 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.13-140.bl.1.4 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.bn.1.2 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.bp.1.2 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.br.1.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.ej.1.13 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.ep.1.13 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.ev.1.7 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.fb.1.7 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.15-140.w.1.7 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.w.1.11 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.x.1.7 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.x.1.11 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.be.1.1 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.be.1.11 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.bf.1.1 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.bf.1.7 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dg.1.3 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dg.1.31 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dj.1.5 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.dj.1.31 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ee.1.5 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ee.1.19 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.eh.1.3 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.eh.1.21 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |