Invariants
Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
Genus: | $5 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $10^{12}$ | Cusp orbits | $4^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $4$ | ||||||
$\overline{\Q}$-gonality: | $4$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 10B5 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.5.79 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&6\\30&9\end{bmatrix}$, $\begin{bmatrix}13&24\\26&9\end{bmatrix}$, $\begin{bmatrix}21&18\\22&23\end{bmatrix}$, $\begin{bmatrix}29&32\\8&17\end{bmatrix}$, $\begin{bmatrix}29&34\\0&1\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.120.5.c.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: | $2^{16}\cdot5^{10}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{5}$ |
Newforms: | 50.2.a.b$^{2}$, 100.2.a.a, 1600.2.a.i$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
$ 0 $ | $=$ | $ 8 x z - 4 y z + 6 z^{2} - 2 w t - t^{2} $ |
$=$ | $8 x^{2} - 2 y^{2} + 6 y z - 2 z^{2} - w^{2} + w t - t^{2}$ | |
$=$ | $8 x y + 6 y^{2} - 4 y z - w^{2} - 2 w t$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 100 x^{4} y^{4} - 100 x^{4} y^{2} z^{2} + 5 x^{4} z^{4} - 3600 x^{2} y^{6} + 2800 x^{2} y^{4} z^{2} + \cdots + z^{8} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=17$, and therefore no 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 :
$\displaystyle X$ | $=$ | $\displaystyle -3x-y-z$ |
$\displaystyle Y$ | $=$ | $\displaystyle x-3y+2z$ |
$\displaystyle Z$ | $=$ | $\displaystyle -x-2y+3z$ |
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 40.120.5.c.1 :
$\displaystyle X$ | $=$ | $\displaystyle x+z$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{4}y$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}w$ |
Equation of the image curve:
$0$ | $=$ | $ 100X^{4}Y^{4}-100X^{4}Y^{2}Z^{2}+5X^{4}Z^{4}-3600X^{2}Y^{6}+2800X^{2}Y^{4}Z^{2}-260X^{2}Y^{2}Z^{4}+32400Y^{8}-36000Y^{6}Z^{2}+10360Y^{4}Z^{4}-200Y^{2}Z^{6}+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.1.2 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
40.120.3-10.a.1.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
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.z.1.1 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
40.480.13-40.z.1.4 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
40.480.13-40.ba.1.3 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{8}$ |
40.480.13-40.ba.1.4 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{8}$ |
40.480.13-40.ba.1.5 | $40$ | $2$ | $2$ | $13$ | $5$ | $1^{8}$ |
40.480.13-40.bc.1.2 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
40.480.13-40.bc.1.12 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
40.480.13-40.bc.1.19 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
40.480.13-40.bd.1.2 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
40.480.13-40.bd.1.3 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
40.480.13-40.bd.1.7 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
40.480.13-40.bl.1.1 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
40.480.13-40.bl.1.4 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
40.480.13-40.bl.1.8 | $40$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
40.480.13-40.bm.1.1 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
40.480.13-40.bm.1.4 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
40.480.13-40.bm.1.7 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
40.480.13-40.bo.1.1 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
40.480.13-40.bo.1.2 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
40.480.13-40.bo.1.5 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
40.480.13-40.bp.1.1 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
40.480.13-40.bp.1.5 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
40.480.13-40.bp.1.8 | $40$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
40.720.13-40.c.1.3 | $40$ | $3$ | $3$ | $13$ | $3$ | $1^{8}$ |
120.480.13-120.cv.1.3 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.cv.1.4 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.cv.1.13 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.cw.1.5 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.cw.1.6 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.cw.1.13 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.cy.1.3 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.cy.1.11 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.cy.1.16 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.cz.1.5 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.cz.1.10 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.cz.1.16 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.ef.1.5 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.ef.1.10 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.ef.1.13 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.eg.1.4 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.eg.1.9 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.eg.1.13 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.ei.1.9 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.ei.1.11 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.ei.1.12 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.ej.1.9 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.ej.1.13 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
120.480.13-120.ej.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.z.1.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.z.1.10 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.z.1.14 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.ba.1.7 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.ba.1.12 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.ba.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bc.1.6 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bc.1.8 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bc.1.16 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bd.1.1 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bd.1.11 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bd.1.15 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bl.1.4 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bl.1.11 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bl.1.15 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bm.1.6 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bm.1.10 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bm.1.14 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bo.1.4 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bo.1.11 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bo.1.14 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bp.1.6 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bp.1.10 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |
280.480.13-280.bp.1.15 | $280$ | $2$ | $2$ | $13$ | $?$ | not computed |