Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
Index: | $96$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $4^{4}$ | ||
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: | 8K1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.1.1007 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&20\\30&21\end{bmatrix}$, $\begin{bmatrix}21&8\\36&19\end{bmatrix}$, $\begin{bmatrix}37&20\\2&21\end{bmatrix}$, $\begin{bmatrix}39&0\\0&31\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 40.192.1-40.d.1.1, 40.192.1-40.d.1.2, 40.192.1-40.d.1.3, 40.192.1-40.d.1.4, 40.192.1-40.d.1.5, 40.192.1-40.d.1.6, 40.192.1-40.d.1.7, 40.192.1-40.d.1.8, 80.192.1-40.d.1.1, 80.192.1-40.d.1.2, 80.192.1-40.d.1.3, 80.192.1-40.d.1.4, 120.192.1-40.d.1.1, 120.192.1-40.d.1.2, 120.192.1-40.d.1.3, 120.192.1-40.d.1.4, 120.192.1-40.d.1.5, 120.192.1-40.d.1.6, 120.192.1-40.d.1.7, 120.192.1-40.d.1.8, 240.192.1-40.d.1.1, 240.192.1-40.d.1.2, 240.192.1-40.d.1.3, 240.192.1-40.d.1.4, 280.192.1-40.d.1.1, 280.192.1-40.d.1.2, 280.192.1-40.d.1.3, 280.192.1-40.d.1.4, 280.192.1-40.d.1.5, 280.192.1-40.d.1.6, 280.192.1-40.d.1.7, 280.192.1-40.d.1.8 |
Cyclic 40-isogeny field degree: | $12$ |
Cyclic 40-torsion field degree: | $192$ |
Full 40-torsion field degree: | $7680$ |
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 y^{2} - z^{2} - w^{2} $ |
$=$ | $10 x^{2} + z^{2} + 2 w^{2}$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{8}+4z^{6}w^{2}+5z^{4}w^{4}+2z^{2}w^{6}+w^{8})^{3}}{w^{8}z^{4}(z^{2}+w^{2})^{4}(z^{2}+2w^{2})^{2}}$ |
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 |
---|---|---|---|---|---|---|
8.48.1.e.2 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.0.m.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.48.0.n.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.48.0.q.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.48.0.r.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.48.1.q.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.48.1.r.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 |
---|---|---|---|---|---|---|
40.480.33.bx.2 | $40$ | $5$ | $5$ | $33$ | $9$ | $1^{14}\cdot2^{9}$ |
40.576.33.hd.2 | $40$ | $6$ | $6$ | $33$ | $5$ | $1^{14}\cdot2\cdot4^{4}$ |
40.960.65.jd.1 | $40$ | $10$ | $10$ | $65$ | $15$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |
80.192.9.gb.2 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
80.192.9.gc.2 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
80.192.9.gg.2 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
80.192.9.gh.2 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.288.17.bkp.2 | $120$ | $3$ | $3$ | $17$ | $?$ | not computed |
120.384.17.lg.2 | $120$ | $4$ | $4$ | $17$ | $?$ | not computed |
240.192.9.rl.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
240.192.9.rm.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
240.192.9.ry.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
240.192.9.rz.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |