Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
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 | $4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8G1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.1.358 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&12\\22&13\end{bmatrix}$, $\begin{bmatrix}15&38\\31&39\end{bmatrix}$, $\begin{bmatrix}17&26\\21&37\end{bmatrix}$, $\begin{bmatrix}39&28\\14&11\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 40.96.1-40.dd.2.1, 40.96.1-40.dd.2.2, 40.96.1-40.dd.2.3, 40.96.1-40.dd.2.4, 80.96.1-40.dd.2.1, 80.96.1-40.dd.2.2, 80.96.1-40.dd.2.3, 80.96.1-40.dd.2.4, 120.96.1-40.dd.2.1, 120.96.1-40.dd.2.2, 120.96.1-40.dd.2.3, 120.96.1-40.dd.2.4, 240.96.1-40.dd.2.1, 240.96.1-40.dd.2.2, 240.96.1-40.dd.2.3, 240.96.1-40.dd.2.4, 280.96.1-40.dd.2.1, 280.96.1-40.dd.2.2, 280.96.1-40.dd.2.3, 280.96.1-40.dd.2.4 |
Cyclic 40-isogeny field degree: | $24$ |
Cyclic 40-torsion field degree: | $384$ |
Full 40-torsion field degree: | $15360$ |
Jacobian
Conductor: | $2^{5}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 10 x^{2} - 10 x y + 5 y^{2} + w^{2} $ |
$=$ | $10 x^{2} + 10 x y + 5 y^{2} - z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} - 10 x^{2} y^{2} + 4 x^{2} z^{2} + 25 y^{4} + 10 y^{2} z^{2} + 2 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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{540y^{2}z^{10}+1620y^{2}z^{8}w^{2}-79560y^{2}z^{6}w^{4}+79560y^{2}z^{4}w^{6}-1620y^{2}z^{2}w^{8}-540y^{2}w^{10}-27z^{12}+486z^{10}w^{2}+6219z^{8}w^{4}-21356z^{6}w^{6}+6219z^{4}w^{8}+486z^{2}w^{10}-27w^{12}}{(z^{2}+w^{2})^{4}(20y^{2}z^{2}-20y^{2}w^{2}-z^{4}-2z^{2}w^{2}-w^{4})}$ |
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.i.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.24.0.ek.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.el.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
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.fq.1 | $40$ | $5$ | $5$ | $17$ | $1$ | $1^{6}\cdot2^{5}$ |
40.288.17.oc.2 | $40$ | $6$ | $6$ | $17$ | $1$ | $1^{6}\cdot2\cdot4^{2}$ |
40.480.33.zm.2 | $40$ | $10$ | $10$ | $33$ | $3$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
80.96.5.dw.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.dx.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.em.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.eo.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.fc.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.fe.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.fr.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.fs.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.9.col.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
120.192.9.xh.2 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
240.96.5.lj.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.ll.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.mx.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.nb.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.od.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.oh.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.px.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.pz.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |