Invariants
| Level: | $7$ | $\SL_2$-level: | $7$ | ||||
| Index: | $8$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $1\cdot7$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-7,-28$) |
Other labels
| Cummins and Pauli (CP) label: | 7B0 |
| Sutherland and Zywina (SZ) label: | 7B0-7a |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 7.8.0.1 |
| Sutherland (S) label: | 7B |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 444 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3^3}{2}\cdot\frac{x^{8}(x^{2}-6xy-12y^{2})^{3}(13x^{2}-30xy+36y^{2})}{x^{15}(2x+3y)}$ |
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 |
|---|---|---|---|---|---|
| $X(1)$ | $1$ | $8$ | $8$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| $X_{\pm1}(7)$ | $7$ | $3$ | $3$ | $0$ |
| 7.24.0.a.2 | $7$ | $3$ | $3$ | $0$ |
| 7.24.0.b.1 | $7$ | $3$ | $3$ | $0$ |
| $X_{\mathrm{sp}}(7)$ | $7$ | $7$ | $7$ | $1$ |
| 14.16.0.a.1 | $14$ | $2$ | $2$ | $0$ |
| 14.16.0.b.1 | $14$ | $2$ | $2$ | $0$ |
| $X_0(14)$ | $14$ | $3$ | $3$ | $1$ |
| 21.24.2.a.1 | $21$ | $3$ | $3$ | $2$ |
| $X_0(21)$ | $21$ | $4$ | $4$ | $1$ |
| 28.16.0.a.1 | $28$ | $2$ | $2$ | $0$ |
| 28.16.0.b.1 | $28$ | $2$ | $2$ | $0$ |
| 28.32.2.a.1 | $28$ | $4$ | $4$ | $2$ |
| 35.40.2.a.1 | $35$ | $5$ | $5$ | $2$ |
| $X_0(35)$ | $35$ | $6$ | $6$ | $3$ |
| 35.80.5.a.1 | $35$ | $10$ | $10$ | $5$ |
| 42.16.0.a.1 | $42$ | $2$ | $2$ | $0$ |
| 42.16.0.b.1 | $42$ | $2$ | $2$ | $0$ |
| $X_0(49)$ | $49$ | $7$ | $7$ | $1$ |
| 56.16.0.a.1 | $56$ | $2$ | $2$ | $0$ |
| 56.16.0.b.1 | $56$ | $2$ | $2$ | $0$ |
| 56.16.0.c.1 | $56$ | $2$ | $2$ | $0$ |
| 56.16.0.d.1 | $56$ | $2$ | $2$ | $0$ |
| 63.24.0.a.1 | $63$ | $3$ | $3$ | $0$ |
| 63.24.0.a.2 | $63$ | $3$ | $3$ | $0$ |
| 63.24.0.b.1 | $63$ | $3$ | $3$ | $0$ |
| 63.24.0.b.2 | $63$ | $3$ | $3$ | $0$ |
| 63.24.0.c.1 | $63$ | $3$ | $3$ | $0$ |
| 63.24.0.c.2 | $63$ | $3$ | $3$ | $0$ |
| 63.216.14.a.1 | $63$ | $27$ | $27$ | $14$ |
| 70.16.0.a.1 | $70$ | $2$ | $2$ | $0$ |
| 70.16.0.b.1 | $70$ | $2$ | $2$ | $0$ |
| $X_0(77)$ | $77$ | $12$ | $12$ | $7$ |
| 84.16.0.c.1 | $84$ | $2$ | $2$ | $0$ |
| 84.16.0.d.1 | $84$ | $2$ | $2$ | $0$ |
| 91.24.0.a.1 | $91$ | $3$ | $3$ | $0$ |
| 91.24.0.a.2 | $91$ | $3$ | $3$ | $0$ |
| 91.24.0.b.1 | $91$ | $3$ | $3$ | $0$ |
| 91.24.0.b.2 | $91$ | $3$ | $3$ | $0$ |
| 91.24.0.c.1 | $91$ | $3$ | $3$ | $0$ |
| 91.24.0.c.2 | $91$ | $3$ | $3$ | $0$ |
| $X_0(91)$ | $91$ | $14$ | $14$ | $7$ |
| $X_0(119)$ | $119$ | $18$ | $18$ | $11$ |
| 133.24.0.a.1 | $133$ | $3$ | $3$ | $0$ |
| 133.24.0.a.2 | $133$ | $3$ | $3$ | $0$ |
| 133.24.0.b.1 | $133$ | $3$ | $3$ | $0$ |
| 133.24.0.b.2 | $133$ | $3$ | $3$ | $0$ |
| 133.24.0.c.1 | $133$ | $3$ | $3$ | $0$ |
| 133.24.0.c.2 | $133$ | $3$ | $3$ | $0$ |
| $X_0(133)$ | $133$ | $20$ | $20$ | $11$ |
| 140.16.0.a.1 | $140$ | $2$ | $2$ | $0$ |
| 140.16.0.b.1 | $140$ | $2$ | $2$ | $0$ |
| 154.16.0.a.1 | $154$ | $2$ | $2$ | $0$ |
| 154.16.0.b.1 | $154$ | $2$ | $2$ | $0$ |
| $X_0(161)$ | $161$ | $24$ | $24$ | $15$ |
| 168.16.0.c.1 | $168$ | $2$ | $2$ | $0$ |
| 168.16.0.d.1 | $168$ | $2$ | $2$ | $0$ |
| 168.16.0.e.1 | $168$ | $2$ | $2$ | $0$ |
| 168.16.0.f.1 | $168$ | $2$ | $2$ | $0$ |
| 182.16.0.a.1 | $182$ | $2$ | $2$ | $0$ |
| 182.16.0.b.1 | $182$ | $2$ | $2$ | $0$ |
| $X_0(203)$ | $203$ | $30$ | $30$ | $19$ |
| 210.16.0.a.1 | $210$ | $2$ | $2$ | $0$ |
| 210.16.0.b.1 | $210$ | $2$ | $2$ | $0$ |
| 217.24.0.a.1 | $217$ | $3$ | $3$ | $0$ |
| 217.24.0.a.2 | $217$ | $3$ | $3$ | $0$ |
| 217.24.0.b.1 | $217$ | $3$ | $3$ | $0$ |
| 217.24.0.b.2 | $217$ | $3$ | $3$ | $0$ |
| 217.24.0.c.1 | $217$ | $3$ | $3$ | $0$ |
| 217.24.0.c.2 | $217$ | $3$ | $3$ | $0$ |
| $X_0(217)$ | $217$ | $32$ | $32$ | $19$ |
| 238.16.0.a.1 | $238$ | $2$ | $2$ | $0$ |
| 238.16.0.b.1 | $238$ | $2$ | $2$ | $0$ |
| 259.24.0.a.1 | $259$ | $3$ | $3$ | $0$ |
| 259.24.0.a.2 | $259$ | $3$ | $3$ | $0$ |
| 259.24.0.b.1 | $259$ | $3$ | $3$ | $0$ |
| 259.24.0.b.2 | $259$ | $3$ | $3$ | $0$ |
| 259.24.0.c.1 | $259$ | $3$ | $3$ | $0$ |
| 259.24.0.c.2 | $259$ | $3$ | $3$ | $0$ |
| $X_0(259)$ | $259$ | $38$ | $38$ | $23$ |
| 266.16.0.a.1 | $266$ | $2$ | $2$ | $0$ |
| 266.16.0.b.1 | $266$ | $2$ | $2$ | $0$ |
| 280.16.0.a.1 | $280$ | $2$ | $2$ | $0$ |
| 280.16.0.b.1 | $280$ | $2$ | $2$ | $0$ |
| 280.16.0.c.1 | $280$ | $2$ | $2$ | $0$ |
| 280.16.0.d.1 | $280$ | $2$ | $2$ | $0$ |
| 301.24.0.a.1 | $301$ | $3$ | $3$ | $0$ |
| 301.24.0.a.2 | $301$ | $3$ | $3$ | $0$ |
| 301.24.0.b.1 | $301$ | $3$ | $3$ | $0$ |
| 301.24.0.b.2 | $301$ | $3$ | $3$ | $0$ |
| 301.24.0.c.1 | $301$ | $3$ | $3$ | $0$ |
| 301.24.0.c.2 | $301$ | $3$ | $3$ | $0$ |
| 308.16.0.a.1 | $308$ | $2$ | $2$ | $0$ |
| 308.16.0.b.1 | $308$ | $2$ | $2$ | $0$ |
| 322.16.0.a.1 | $322$ | $2$ | $2$ | $0$ |
| 322.16.0.b.1 | $322$ | $2$ | $2$ | $0$ |