Invariants
| Level: | $20$ | $\SL_2$-level: | $10$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot10$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.12.0.22 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}3&2\\0&17\end{bmatrix}$, $\begin{bmatrix}15&14\\9&1\end{bmatrix}$, $\begin{bmatrix}19&9\\15&2\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 20-isogeny field degree: | $6$ |
| Cyclic 20-torsion field degree: | $48$ |
| Full 20-torsion field degree: | $3840$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 11 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^4}\cdot\frac{x^{12}(5x^{4}-160x^{2}y^{2}+256y^{4})^{3}}{y^{2}x^{22}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 20.24.1.a.1 | $20$ | $2$ | $2$ | $1$ |
| 20.24.1.b.1 | $20$ | $2$ | $2$ | $1$ |
| 20.24.1.d.2 | $20$ | $2$ | $2$ | $1$ |
| 20.24.1.e.2 | $20$ | $2$ | $2$ | $1$ |
| 20.36.0.d.2 | $20$ | $3$ | $3$ | $0$ |
| 20.48.1.a.1 | $20$ | $4$ | $4$ | $1$ |
| 20.60.2.c.1 | $20$ | $5$ | $5$ | $2$ |
| 40.24.1.cb.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.ce.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.cn.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.cq.2 | $40$ | $2$ | $2$ | $1$ |
| 60.24.1.s.1 | $60$ | $2$ | $2$ | $1$ |
| 60.24.1.t.2 | $60$ | $2$ | $2$ | $1$ |
| 60.24.1.bf.1 | $60$ | $2$ | $2$ | $1$ |
| 60.24.1.bg.1 | $60$ | $2$ | $2$ | $1$ |
| 60.36.0.ch.2 | $60$ | $3$ | $3$ | $0$ |
| 60.48.3.bd.1 | $60$ | $4$ | $4$ | $3$ |
| 100.60.2.a.2 | $100$ | $5$ | $5$ | $2$ |
| 120.24.1.he.2 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.hk.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.jy.2 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ke.2 | $120$ | $2$ | $2$ | $1$ |
| 140.24.1.m.1 | $140$ | $2$ | $2$ | $1$ |
| 140.24.1.n.1 | $140$ | $2$ | $2$ | $1$ |
| 140.24.1.p.2 | $140$ | $2$ | $2$ | $1$ |
| 140.24.1.q.2 | $140$ | $2$ | $2$ | $1$ |
| 140.96.7.f.1 | $140$ | $8$ | $8$ | $7$ |
| 140.252.14.b.1 | $140$ | $21$ | $21$ | $14$ |
| 140.336.21.if.2 | $140$ | $28$ | $28$ | $21$ |
| 180.324.22.ib.1 | $180$ | $27$ | $27$ | $22$ |
| 220.24.1.m.1 | $220$ | $2$ | $2$ | $1$ |
| 220.24.1.n.1 | $220$ | $2$ | $2$ | $1$ |
| 220.24.1.p.2 | $220$ | $2$ | $2$ | $1$ |
| 220.24.1.q.2 | $220$ | $2$ | $2$ | $1$ |
| 220.144.11.f.2 | $220$ | $12$ | $12$ | $11$ |
| 260.24.1.m.2 | $260$ | $2$ | $2$ | $1$ |
| 260.24.1.n.2 | $260$ | $2$ | $2$ | $1$ |
| 260.24.1.p.1 | $260$ | $2$ | $2$ | $1$ |
| 260.24.1.q.1 | $260$ | $2$ | $2$ | $1$ |
| 260.168.11.fd.1 | $260$ | $14$ | $14$ | $11$ |
| 280.24.1.le.2 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.lh.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.lq.2 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.lt.2 | $280$ | $2$ | $2$ | $1$ |