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: | $2$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20C7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.7.607 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&0\\12&7\end{bmatrix}$, $\begin{bmatrix}17&38\\26&3\end{bmatrix}$, $\begin{bmatrix}37&38\\6&23\end{bmatrix}$, $\begin{bmatrix}39&14\\6&5\end{bmatrix}$, $\begin{bmatrix}39&30\\0&39\end{bmatrix}$, $\begin{bmatrix}39&32\\18&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.120.7.a.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.c, 400.2.a.e, 400.2.a.f |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x^{2} - 2 x y + x z + y^{2} - y z - z^{2} + t^{2} + t v - v^{2} $ |
| $=$ | $x z - y^{2} + 2 y z - w^{2} - w u + w v - 2 t^{2} + t v + u^{2}$ | |
| $=$ | $3 x z + 2 y^{2} + y z + w v + t^{2} - u v$ | |
| $=$ | $x^{2} - x y - 2 y^{2} - y z - w t + t u - t v + u v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 9 x^{8} y^{2} + 48 x^{7} y^{3} + 36 x^{7} y z^{2} - 94 x^{6} y^{4} - 98 x^{6} y^{2} z^{2} + \cdots + 4 y^{2} 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 :
| $\displaystyle X$ | $=$ | $\displaystyle 2x+3y+z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -4x-y-2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x+y-3z$ |
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.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ -9X^{8}Y^{2}+48X^{7}Y^{3}+36X^{7}YZ^{2}-94X^{6}Y^{4}-98X^{6}Y^{2}Z^{2}-36X^{6}Z^{4}+44X^{5}Y^{5}+49X^{5}Y^{3}Z^{2}-4X^{5}YZ^{4}+119X^{4}Y^{6}+120X^{4}Y^{4}Z^{2}+111X^{4}Y^{2}Z^{4}+16X^{4}Z^{6}-188X^{3}Y^{7}-115X^{3}Y^{5}Z^{2}-83X^{3}Y^{3}Z^{4}-24X^{3}YZ^{6}+44X^{2}Y^{8}-164X^{2}Y^{6}Z^{2}-190X^{2}Y^{4}Z^{4}-104X^{2}Y^{2}Z^{6}-4X^{2}Z^{8}+96XY^{9}+228XY^{7}Z^{2}+145XY^{5}Z^{4}-24XY^{3}Z^{6}-4XYZ^{8}-64Y^{10}-20Y^{8}Z^{2}+29Y^{6}Z^{4}+60Y^{4}Z^{6}+4Y^{2}Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.24.0-4.a.1.4 | $40$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 40.120.3-10.a.1.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.120.3-10.a.1.8 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
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.a.1.1 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.13-40.b.1.2 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-20.c.1.1 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
| 40.480.13-40.h.1.2 | $40$ | $2$ | $2$ | $13$ | $6$ | $1^{6}$ |
| 40.480.13-20.i.1.2 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.13-20.k.1.1 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 40.480.13-40.z.1.4 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
| 40.480.13-40.bf.1.4 | $40$ | $2$ | $2$ | $13$ | $6$ | $1^{6}$ |
| 40.480.15-20.a.1.6 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.a.1.2 | $40$ | $2$ | $2$ | $15$ | $10$ | $1^{8}$ |
| 40.480.15-20.b.1.14 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.b.1.2 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-20.c.1.6 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-20.d.1.5 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.d.1.1 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-20.f.1.3 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.f.1.1 | $40$ | $2$ | $2$ | $15$ | $8$ | $1^{8}$ |
| 40.480.15-20.g.1.3 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-20.i.1.4 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-20.j.1.4 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.k.1.15 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.l.1.15 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.o.1.14 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.15-40.p.1.16 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.ba.1.6 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.bc.1.6 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.bh.1.8 | $40$ | $2$ | $2$ | $15$ | $10$ | $1^{8}$ |
| 40.480.15-40.bj.1.8 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 40.480.15-40.ca.1.16 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.cb.1.14 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 40.480.15-40.ce.1.15 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 40.480.15-40.cf.1.15 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 40.480.17-40.m.1.14 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{6}\cdot2^{2}$ |
| 40.480.17-40.n.1.16 | $40$ | $2$ | $2$ | $17$ | $6$ | $1^{6}\cdot2^{2}$ |
| 40.480.17-40.ce.1.15 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{6}\cdot2^{2}$ |
| 40.480.17-40.cf.1.15 | $40$ | $2$ | $2$ | $17$ | $4$ | $1^{6}\cdot2^{2}$ |
| 40.480.17-40.cu.1.15 | $40$ | $2$ | $2$ | $17$ | $6$ | $1^{4}\cdot2^{3}$ |
| 40.480.17-40.cv.1.15 | $40$ | $2$ | $2$ | $17$ | $6$ | $1^{4}\cdot2^{3}$ |
| 40.480.17-40.cy.1.16 | $40$ | $2$ | $2$ | $17$ | $6$ | $1^{4}\cdot2^{3}$ |
| 40.480.17-40.cz.1.14 | $40$ | $2$ | $2$ | $17$ | $6$ | $1^{4}\cdot2^{3}$ |
| 40.720.19-20.i.1.4 | $40$ | $3$ | $3$ | $19$ | $4$ | $1^{12}$ |
| 120.480.13-60.e.1.6 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.g.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.n.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.t.1.12 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.bc.1.5 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-60.be.1.3 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.dh.1.6 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.dn.1.6 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.15-60.a.1.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.a.1.2 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.b.1.6 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.c.1.2 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.d.1.5 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.e.1.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.h.1.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.j.1.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.k.1.2 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.l.1.2 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.r.1.2 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-60.s.1.2 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ba.1.29 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bb.1.29 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.be.1.25 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bf.1.25 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bq.1.14 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bs.1.4 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cj.1.4 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cl.1.6 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.em.1.25 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.en.1.25 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.eq.1.29 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.er.1.29 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.17-120.ds.1.25 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.dt.1.25 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.gm.1.29 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.gn.1.29 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.ia.1.29 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.ib.1.29 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.ie.1.25 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.if.1.25 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.13-140.q.1.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.s.1.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.y.1.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-140.ba.1.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.bx.1.9 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.cd.1.5 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.cv.1.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.13-280.db.1.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 280.480.15-140.a.1.17 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.a.1.11 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.b.1.19 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.c.1.11 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.d.1.11 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.e.1.1 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.h.1.1 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.j.1.1 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.k.1.6 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.l.1.2 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.n.1.2 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-140.o.1.6 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.ba.1.30 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.bb.1.26 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.be.1.24 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.bf.1.22 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.bq.1.2 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.bs.1.2 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.bx.1.4 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.bz.1.6 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cq.1.30 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cr.1.28 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cu.1.22 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.15-280.cv.1.24 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 280.480.17-280.fo.1.28 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.fp.1.30 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.fs.1.24 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.ft.1.22 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.hs.1.26 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.ht.1.30 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.hw.1.22 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.hx.1.24 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |