Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8F1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.1.393 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&30\\37&21\end{bmatrix}$, $\begin{bmatrix}9&8\\14&37\end{bmatrix}$, $\begin{bmatrix}25&16\\7&23\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 80.96.1-40.ex.1.1, 80.96.1-40.ex.1.2, 80.96.1-40.ex.1.3, 80.96.1-40.ex.1.4, 240.96.1-40.ex.1.1, 240.96.1-40.ex.1.2, 240.96.1-40.ex.1.3, 240.96.1-40.ex.1.4 |
Cyclic 40-isogeny field degree: | $12$ |
Cyclic 40-torsion field degree: | $192$ |
Full 40-torsion field degree: | $15360$ |
Jacobian
Conductor: | $2^{6}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 y^{2} + 2 y z + 2 z^{2} + 2 w^{2} $ |
$=$ | $40 x^{2} - y^{2} + y z + z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} + 8 x^{2} y^{2} + 30 x^{2} z^{2} + 9 y^{4} + 30 y^{2} z^{2} + 25 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 4x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{2^6}{3^4}\cdot\frac{4812500yz^{11}+14437500yz^{9}w^{2}+11340000yz^{7}w^{4}-378000yz^{5}w^{6}-26697600yz^{3}w^{8}-14929920yzw^{10}+359375z^{12}+331250z^{10}w^{2}-16267500z^{8}w^{4}-37485000z^{6}w^{6}-33134400z^{4}w^{8}-8709120z^{2}w^{10}+7962624w^{12}}{w^{4}(5000yz^{7}+9000yz^{5}w^{2}-3600yz^{3}w^{4}-4320yzw^{6}+18125z^{8}+42500z^{6}w^{2}+29100z^{4}w^{4}+6480z^{2}w^{6}-432w^{8})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
8.24.1.t.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.24.0.cd.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.cl.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.dl.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.dz.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.1.y.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.24.1.bp.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
40.240.17.jb.1 | $40$ | $5$ | $5$ | $17$ | $5$ | $1^{14}\cdot2$ |
40.288.17.xh.1 | $40$ | $6$ | $6$ | $17$ | $4$ | $1^{14}\cdot2$ |
40.480.33.bov.1 | $40$ | $10$ | $10$ | $33$ | $7$ | $1^{28}\cdot2^{2}$ |
120.144.9.ezb.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
120.192.9.box.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |