Invariants
Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
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: | $4$ | ||||||
$\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
$\overline{\Q}$-gonality: | $4$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 20C7 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.7.238 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&28\\12&19\end{bmatrix}$, $\begin{bmatrix}13&38\\12&25\end{bmatrix}$, $\begin{bmatrix}29&32\\28&7\end{bmatrix}$, $\begin{bmatrix}33&10\\30&23\end{bmatrix}$, $\begin{bmatrix}37&2\\30&3\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.120.7.a.1 for the level structure with $-I$) |
Cyclic 40-isogeny field degree: | $24$ |
Cyclic 40-torsion field degree: | $192$ |
Full 40-torsion field degree: | $3072$ |
Jacobian
Conductor: | $2^{28}\cdot5^{14}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{7}$ |
Newforms: | 50.2.a.b$^{2}$, 100.2.a.a, 1600.2.a.a, 1600.2.a.c, 1600.2.a.o, 1600.2.a.q |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
$ 0 $ | $=$ | $ x t - x u - x v - y w - y t - y u - 2 y v - z w + z t $ |
$=$ | $x w + x u - 2 y t - y u - 2 y v + z w + z t + z u + 2 z v$ | |
$=$ | $2 x^{2} + 2 y^{2} + 2 y z - 2 z^{2} + w t + w v + u v + v^{2}$ | |
$=$ | $2 x^{2} - 2 x z - 2 y z - 4 z^{2} - w u + t^{2} - t u + t v - u^{2} - u v$ | |
$=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 256 x^{10} - 384 x^{9} y - 176 x^{8} y^{2} - 40 x^{8} z^{2} + 752 x^{7} y^{3} + 456 x^{7} y z^{2} + \cdots + 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 :
$\displaystyle X$ | $=$ | $\displaystyle -2x-y-3z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 4x+2y+z$ |
$\displaystyle Z$ | $=$ | $\displaystyle x+3y-z$ |
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 40.120.7.a.1 :
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle z$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
$0$ | $=$ | $ 256X^{10}-384X^{9}Y-176X^{8}Y^{2}-40X^{8}Z^{2}+752X^{7}Y^{3}+456X^{7}YZ^{2}-476X^{6}Y^{4}-328X^{6}Y^{2}Z^{2}-29X^{6}Z^{4}-176X^{5}Y^{5}-230X^{5}Y^{3}Z^{2}-145X^{5}YZ^{4}+376X^{4}Y^{6}+240X^{4}Y^{4}Z^{2}+190X^{4}Y^{2}Z^{4}+30X^{4}Z^{6}-192X^{3}Y^{7}+98X^{3}Y^{5}Z^{2}+83X^{3}Y^{3}Z^{4}-12X^{3}YZ^{6}+36X^{2}Y^{8}-196X^{2}Y^{6}Z^{2}-111X^{2}Y^{4}Z^{4}-52X^{2}Y^{2}Z^{6}-X^{2}Z^{8}+72XY^{7}Z^{2}+4XY^{5}Z^{4}-12XY^{3}Z^{6}+XYZ^{8}+36Y^{6}Z^{4}+8Y^{4}Z^{6}+Y^{2}Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
20.120.3-10.a.1.3 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
40.24.0-8.a.1.2 | $40$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
40.120.3-10.a.1.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
40.480.13-40.a.1.3 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{6}$ |
40.480.13-40.c.1.8 | $40$ | $2$ | $2$ | $13$ | $7$ | $1^{6}$ |
40.480.13-40.g.1.5 | $40$ | $2$ | $2$ | $13$ | $10$ | $1^{6}$ |
40.480.13-40.i.1.3 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
40.480.13-40.y.1.2 | $40$ | $2$ | $2$ | $13$ | $6$ | $1^{6}$ |
40.480.13-40.ba.1.5 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
40.480.13-40.be.1.5 | $40$ | $2$ | $2$ | $13$ | $6$ | $1^{6}$ |
40.480.13-40.bg.1.7 | $40$ | $2$ | $2$ | $13$ | $7$ | $1^{6}$ |
40.480.15-40.a.1.2 | $40$ | $2$ | $2$ | $15$ | $10$ | $1^{8}$ |
40.480.15-40.a.1.3 | $40$ | $2$ | $2$ | $15$ | $10$ | $1^{8}$ |
40.480.15-40.c.1.6 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
40.480.15-40.c.1.12 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
40.480.15-40.e.1.4 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
40.480.15-40.e.1.8 | $40$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
40.480.15-40.g.1.7 | $40$ | $2$ | $2$ | $15$ | $7$ | $1^{8}$ |
40.480.15-40.g.1.8 | $40$ | $2$ | $2$ | $15$ | $7$ | $1^{8}$ |
40.480.15-40.bb.1.7 | $40$ | $2$ | $2$ | $15$ | $7$ | $1^{8}$ |
40.480.15-40.bb.1.8 | $40$ | $2$ | $2$ | $15$ | $7$ | $1^{8}$ |
40.480.15-40.bd.1.7 | $40$ | $2$ | $2$ | $15$ | $7$ | $1^{8}$ |
40.480.15-40.bd.1.8 | $40$ | $2$ | $2$ | $15$ | $7$ | $1^{8}$ |
40.480.15-40.bi.1.5 | $40$ | $2$ | $2$ | $15$ | $9$ | $1^{8}$ |
40.480.15-40.bi.1.8 | $40$ | $2$ | $2$ | $15$ | $9$ | $1^{8}$ |
40.480.15-40.bk.1.5 | $40$ | $2$ | $2$ | $15$ | $5$ | $1^{8}$ |
40.480.15-40.bk.1.8 | $40$ | $2$ | $2$ | $15$ | $5$ | $1^{8}$ |
40.720.19-40.z.1.6 | $40$ | $3$ | $3$ | $19$ | $9$ | $1^{12}$ |
120.480.13-120.m.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.o.1.13 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.s.1.11 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.u.1.11 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.dg.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.di.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.dm.1.11 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.do.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.15-120.b.1.8 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.b.1.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.d.1.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.d.1.32 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.i.1.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.i.1.24 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.k.1.6 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.k.1.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.br.1.15 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.br.1.25 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.bt.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.bt.1.23 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.ck.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.ck.1.15 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.cm.1.15 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
120.480.15-120.cm.1.29 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.13-280.bw.1.8 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.by.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.cc.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.ce.1.8 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.cu.1.15 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.cw.1.10 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.da.1.10 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.dc.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.15-280.b.1.7 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.b.1.17 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.d.1.1 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.d.1.21 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.i.1.9 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.i.1.27 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.k.1.29 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.k.1.31 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.br.1.3 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.br.1.15 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.bt.1.3 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.bt.1.7 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.by.1.7 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.by.1.13 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.ca.1.7 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |
280.480.15-280.ca.1.15 | $280$ | $2$ | $2$ | $15$ | $?$ | not computed |