Invariants
Level: | $63$ | $\SL_2$-level: | $9$ | ||||
Index: | $18$ | $\PSL_2$-index: | $18$ | ||||
Genus: | $0 = 1 + \frac{ 18 }{12} - \frac{ 6 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
Cusps: | $2$ (none of which are rational) | Cusp widths | $9^{2}$ | Cusp orbits | $2$ | ||
Elliptic points: | $6$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 9D0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 63.18.0.1 |
Level structure
$\GL_2(\Z/63\Z)$-generators: | $\begin{bmatrix}50&3\\18&17\end{bmatrix}$, $\begin{bmatrix}61&20\\49&13\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 63-isogeny field degree: | $96$ |
Cyclic 63-torsion field degree: | $3456$ |
Full 63-torsion field degree: | $435456$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 7 x^{2} - 7 x y + 7 y^{2} + 9 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
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 |
---|---|---|---|---|---|
$X_{\mathrm{ns}}(3)$ | $3$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
63.144.11.g.1 | $63$ | $8$ | $8$ | $11$ |
63.162.4.b.1 | $63$ | $9$ | $9$ | $4$ |
63.378.22.b.1 | $63$ | $21$ | $21$ | $22$ |
63.504.33.c.1 | $63$ | $28$ | $28$ | $33$ |
126.36.3.e.1 | $126$ | $2$ | $2$ | $3$ |
126.36.3.j.1 | $126$ | $2$ | $2$ | $3$ |
126.36.3.o.1 | $126$ | $2$ | $2$ | $3$ |
126.36.3.r.1 | $126$ | $2$ | $2$ | $3$ |
126.54.2.d.1 | $126$ | $3$ | $3$ | $2$ |
252.36.3.b.1 | $252$ | $2$ | $2$ | $3$ |
252.36.3.k.1 | $252$ | $2$ | $2$ | $3$ |
252.36.3.t.1 | $252$ | $2$ | $2$ | $3$ |
252.36.3.w.1 | $252$ | $2$ | $2$ | $3$ |
252.72.3.cv.1 | $252$ | $4$ | $4$ | $3$ |
315.90.6.b.1 | $315$ | $5$ | $5$ | $6$ |
315.108.5.b.1 | $315$ | $6$ | $6$ | $5$ |
315.180.11.e.1 | $315$ | $10$ | $10$ | $11$ |