Invariants
| Level: | $15$ | $\SL_2$-level: | $5$ | ||||
| Index: | $20$ | $\PSL_2$-index: | $20$ | ||||
| Genus: | $0 = 1 + \frac{ 20 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $5^{4}$ | Cusp orbits | $4$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3$) |
Other labels
| Cummins and Pauli (CP) label: | 5F0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 15.20.0.2 |
Level structure
| $\GL_2(\Z/15\Z)$-generators: | $\begin{bmatrix}4&12\\11&1\end{bmatrix}$, $\begin{bmatrix}6&4\\7&14\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 15-isogeny field degree: | $24$ |
| Cyclic 15-torsion field degree: | $192$ |
| Full 15-torsion field degree: | $1152$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 9 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 20 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^{12}\cdot3\cdot5^4\cdot11\cdot19\,\frac{(19x-35y)^{3}(19x-2y)^{20}(38x+29y)^{3}(19x^{2}-10xy-20y^{2})(10295359x^{4}+15809995x^{3}y+9654945x^{2}y^{2}+6953050xy^{3}+3885100y^{4})^{3}}{(19x-2y)^{20}(11598569x^{4}+36764240x^{3}y-2945760x^{2}y^{2}-38893000xy^{3}-14892400y^{4})^{5}}$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 15.60.0.b.1 | $15$ | $3$ | $3$ | $0$ |
| 15.60.4.f.1 | $15$ | $3$ | $3$ | $4$ |
| 15.80.3.d.1 | $15$ | $4$ | $4$ | $3$ |
| 30.40.1.b.1 | $30$ | $2$ | $2$ | $1$ |
| 30.40.1.e.1 | $30$ | $2$ | $2$ | $1$ |
| 30.40.1.j.1 | $30$ | $2$ | $2$ | $1$ |
| 30.40.1.l.1 | $30$ | $2$ | $2$ | $1$ |
| 30.60.2.b.1 | $30$ | $3$ | $3$ | $2$ |
| 45.540.38.e.1 | $45$ | $27$ | $27$ | $38$ |
| 60.40.1.e.1 | $60$ | $2$ | $2$ | $1$ |
| 60.40.1.n.1 | $60$ | $2$ | $2$ | $1$ |
| 60.40.1.bc.1 | $60$ | $2$ | $2$ | $1$ |
| 60.40.1.bi.1 | $60$ | $2$ | $2$ | $1$ |
| 60.80.5.o.1 | $60$ | $4$ | $4$ | $5$ |
| 75.100.4.a.1 | $75$ | $5$ | $5$ | $4$ |
| 105.160.9.b.1 | $105$ | $8$ | $8$ | $9$ |
| 120.40.1.n.1 | $120$ | $2$ | $2$ | $1$ |
| 120.40.1.t.1 | $120$ | $2$ | $2$ | $1$ |
| 120.40.1.ca.1 | $120$ | $2$ | $2$ | $1$ |
| 120.40.1.cd.1 | $120$ | $2$ | $2$ | $1$ |
| 120.40.1.ei.1 | $120$ | $2$ | $2$ | $1$ |
| 120.40.1.el.1 | $120$ | $2$ | $2$ | $1$ |
| 120.40.1.fg.1 | $120$ | $2$ | $2$ | $1$ |
| 120.40.1.fj.1 | $120$ | $2$ | $2$ | $1$ |
| 165.240.17.b.1 | $165$ | $12$ | $12$ | $17$ |
| 195.280.19.b.1 | $195$ | $14$ | $14$ | $19$ |
| 210.40.1.q.1 | $210$ | $2$ | $2$ | $1$ |
| 210.40.1.r.1 | $210$ | $2$ | $2$ | $1$ |
| 210.40.1.w.1 | $210$ | $2$ | $2$ | $1$ |
| 210.40.1.x.1 | $210$ | $2$ | $2$ | $1$ |
| 330.40.1.q.1 | $330$ | $2$ | $2$ | $1$ |
| 330.40.1.r.1 | $330$ | $2$ | $2$ | $1$ |
| 330.40.1.w.1 | $330$ | $2$ | $2$ | $1$ |
| 330.40.1.x.1 | $330$ | $2$ | $2$ | $1$ |