Invariants
Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $200$ | ||
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: | 40.240.7.713 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&36\\14&21\end{bmatrix}$, $\begin{bmatrix}9&14\\16&15\end{bmatrix}$, $\begin{bmatrix}13&30\\22&7\end{bmatrix}$, $\begin{bmatrix}25&38\\2&17\end{bmatrix}$, $\begin{bmatrix}35&38\\32&17\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 20.120.7.g.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: | $2^{14}\cdot5^{14}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{7}$ |
Newforms: | 50.2.a.a, 50.2.a.b$^{2}$, 100.2.a.a, 200.2.a.a, 200.2.a.b, 200.2.a.d |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
$ 0 $ | $=$ | $ x w - x t + x v + y t - y u - y v + 2 z w - z t $ |
$=$ | $2 x w - x t - y t + y u + y v - 2 z w + z t - z u + z v$ | |
$=$ | $x w + x t + 2 x u - y u + y v + z u - z v$ | |
$=$ | $x w - x v - 3 y w + 2 y u + y v + z t + z u + 2 z v$ | |
$=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 25 x^{12} - 5050 x^{10} y^{2} + 350 x^{10} z^{2} + 239625 x^{8} y^{4} + 40000 x^{8} y^{2} z^{2} + \cdots + 13456 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.g.1 :
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y+\frac{1}{2}z$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
$0$ | $=$ | $ 25X^{12}-5050X^{10}Y^{2}+350X^{10}Z^{2}+239625X^{8}Y^{4}+40000X^{8}Y^{2}Z^{2}+935X^{8}Z^{4}+1556000X^{6}Y^{6}-21750X^{6}Y^{4}Z^{2}-21360X^{6}Y^{2}Z^{4}-2030X^{6}Z^{6}+2311000X^{4}Y^{8}+4370000X^{4}Y^{6}Z^{2}+468145X^{4}Y^{4}Z^{4}+23070X^{4}Y^{2}Z^{6}+841X^{4}Z^{8}-184800X^{2}Y^{10}+273000X^{2}Y^{8}Z^{2}+956820X^{2}Y^{6}Z^{4}-129760X^{2}Y^{4}Z^{6}-23548X^{2}Y^{2}Z^{8}+3600Y^{12}-3200Y^{10}Z^{2}+47200Y^{8}Z^{4}+132480Y^{6}Z^{6}+13456Y^{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.1.4 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
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.b.1.12 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
40.480.13-20.c.1.4 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
40.480.13-40.f.1.3 | $40$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
40.480.13-40.i.1.7 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
40.480.13-20.n.1.3 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
40.480.13-20.o.1.3 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
40.480.13-40.bp.1.5 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
40.480.13-40.bs.1.1 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
40.480.15-20.i.1.2 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
40.480.15-20.i.1.4 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
40.480.15-20.k.1.6 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
40.480.15-20.k.1.10 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
40.480.15-20.l.1.2 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
40.480.15-20.l.1.6 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
40.480.15-20.n.1.3 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
40.480.15-20.n.1.7 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
40.480.15-40.bi.1.5 | $40$ | $2$ | $2$ | $15$ | $9$ | $1^{8}$ |
40.480.15-40.bi.1.13 | $40$ | $2$ | $2$ | $15$ | $9$ | $1^{8}$ |
40.480.15-40.bm.1.3 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
40.480.15-40.bm.1.11 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
40.480.15-40.bq.1.4 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
40.480.15-40.bq.1.12 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
40.480.15-40.bw.1.6 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
40.480.15-40.bw.1.14 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
40.720.19-20.s.1.3 | $40$ | $3$ | $3$ | $19$ | $2$ | $1^{12}$ |
120.480.13-60.j.1.3 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-60.k.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.bd.1.5 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.bg.1.5 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-60.bt.1.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-60.bu.1.4 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.fh.1.11 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.fk.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.15-60.u.1.15 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-60.u.1.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-60.w.1.4 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-60.w.1.11 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-60.bg.1.6 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-60.bg.1.13 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-60.bi.1.15 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-60.bi.1.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.cs.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.cs.1.8 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.cy.1.11 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.cy.1.28 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.ec.1.5 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.ec.1.22 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.ei.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.ei.1.4 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.13-140.bg.1.8 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-140.bi.1.4 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-140.bs.1.6 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-140.bu.1.4 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.du.1.8 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.ea.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.fe.1.6 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.fk.1.10 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.15-140.y.1.7 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-140.y.1.14 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-140.z.1.10 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-140.z.1.15 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-140.bg.1.8 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-140.bg.1.9 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-140.bh.1.11 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-140.bh.1.16 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.dm.1.15 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.dm.1.22 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.dp.1.10 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.dp.1.21 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.ek.1.10 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.ek.1.21 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.en.1.3 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.en.1.32 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |