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: | 40.48.1.377 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}17&26\\17&35\end{bmatrix}$, $\begin{bmatrix}19&38\\0&9\end{bmatrix}$, $\begin{bmatrix}25&12\\33&35\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 80.96.1-40.ez.1.1, 80.96.1-40.ez.1.2, 80.96.1-40.ez.1.3, 80.96.1-40.ez.1.4, 240.96.1-40.ez.1.1, 240.96.1-40.ez.1.2, 240.96.1-40.ez.1.3, 240.96.1-40.ez.1.4 |
Cyclic 40-isogeny field degree: | $24$ |
Cyclic 40-torsion field degree: | $384$ |
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 $ | $=$ | $ 5 x^{2} + 2 y^{2} + 3 y z + 3 z^{2} + w^{2} $ |
$=$ | $5 x^{2} - 5 y^{2} - 5 y z - 5 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 6 x^{2} y^{2} + 20 x^{2} z^{2} + 49 y^{4} + 210 y^{2} z^{2} + 225 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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{233766000000yz^{11}-1167075000000yz^{9}w^{2}-4842127080000yz^{7}w^{4}-3116065820000yz^{5}w^{6}+591905564600yz^{3}w^{8}+515529514500yzw^{10}+535221000000z^{12}+1406295000000z^{10}w^{2}-1881774090000z^{8}w^{4}-4862237660000z^{6}w^{6}-2202519174100z^{4}w^{8}-87861473700z^{2}w^{10}+49147147049w^{12}}{2164500000yz^{11}+917000000yz^{9}w^{2}+959420000yz^{7}w^{4}+482601000yz^{5}w^{6}+152103350yz^{3}w^{8}-5546310yzw^{10}+4955750000z^{12}+5605100000z^{10}w^{2}+3307535000z^{8}w^{4}+1037575000z^{6}w^{6}+138717775z^{4}w^{8}-25834760z^{2}w^{10}+151263w^{12}}$ |
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.r.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.24.0.ce.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.co.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.df.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.dv.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.1.z.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.jd.1 | $40$ | $5$ | $5$ | $17$ | $6$ | $1^{14}\cdot2$ |
40.288.17.xj.1 | $40$ | $6$ | $6$ | $17$ | $4$ | $1^{14}\cdot2$ |
40.480.33.box.1 | $40$ | $10$ | $10$ | $33$ | $10$ | $1^{28}\cdot2^{2}$ |
120.144.9.ezd.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
120.192.9.boz.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |