Invariants
| Level: | $7$ | $\SL_2$-level: | $7$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $1^{3}\cdot7^{3}$ | Cusp orbits | $3^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-7,-28$) |
Other labels
| Cummins and Pauli (CP) label: | 7E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 7.48.0.6 |
| Sutherland (S) label: | 7B.1.5 |
Level structure
| $\GL_2(\Z/7\Z)$-generators: | $\begin{bmatrix}3&1\\0&4\end{bmatrix}$, $\begin{bmatrix}3&6\\0&4\end{bmatrix}$ |
| $\GL_2(\Z/7\Z)$-subgroup: | $C_3\times D_7$ |
| Contains $-I$: | no $\quad$ (see 7.24.0.b.1 for the level structure with $-I$) |
| Cyclic 7-isogeny field degree: | $1$ |
| Cyclic 7-torsion field degree: | $6$ |
| Full 7-torsion field degree: | $42$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 20 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -7^5\,\frac{(x-4y)^{24}(4x^{2}+78xy-255y^{2})^{3}(12x^{2}-30xy+49y^{2})^{3}(20x^{2}+82xy-43y^{2})^{3}(68x^{2}-214xy+131y^{2})^{3}}{(x-4y)^{24}(104x^{3}-192x^{2}y-794xy^{2}+923y^{3})^{7}(568x^{3}+444x^{2}y-8068xy^{2}+6119y^{3})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 7.16.0-7.a.1.1 | $7$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 7.336.3-7.a.1.1 | $7$ | $7$ | $7$ | $3$ |
| 14.96.2-14.d.1.2 | $14$ | $2$ | $2$ | $2$ |
| 14.96.2-14.g.1.2 | $14$ | $2$ | $2$ | $2$ |
| 14.144.1-14.b.1.1 | $14$ | $3$ | $3$ | $1$ |
| 21.144.4-21.b.1.1 | $21$ | $3$ | $3$ | $4$ |
| 21.192.3-21.b.1.3 | $21$ | $4$ | $4$ | $3$ |
| 28.96.2-28.e.1.4 | $28$ | $2$ | $2$ | $2$ |
| 28.96.2-28.j.1.2 | $28$ | $2$ | $2$ | $2$ |
| 28.192.6-28.l.1.4 | $28$ | $4$ | $4$ | $6$ |
| 35.240.8-35.b.1.4 | $35$ | $5$ | $5$ | $8$ |
| 35.288.7-35.b.1.2 | $35$ | $6$ | $6$ | $7$ |
| 35.480.15-35.b.1.3 | $35$ | $10$ | $10$ | $15$ |
| 42.96.2-42.b.1.4 | $42$ | $2$ | $2$ | $2$ |
| 42.96.2-42.g.1.4 | $42$ | $2$ | $2$ | $2$ |
| 49.336.3-49.a.1.1 | $49$ | $7$ | $7$ | $3$ |
| 49.336.12-49.a.1.1 | $49$ | $7$ | $7$ | $12$ |
| 49.336.12-49.b.1.1 | $49$ | $7$ | $7$ | $12$ |
| 56.96.2-56.h.1.2 | $56$ | $2$ | $2$ | $2$ |
| 56.96.2-56.i.1.2 | $56$ | $2$ | $2$ | $2$ |
| 56.96.2-56.p.1.2 | $56$ | $2$ | $2$ | $2$ |
| 56.96.2-56.q.1.2 | $56$ | $2$ | $2$ | $2$ |
| 63.1296.46-63.d.1.1 | $63$ | $27$ | $27$ | $46$ |
| 70.96.2-70.b.1.1 | $70$ | $2$ | $2$ | $2$ |
| 70.96.2-70.f.1.1 | $70$ | $2$ | $2$ | $2$ |
| 84.96.2-84.h.1.7 | $84$ | $2$ | $2$ | $2$ |
| 84.96.2-84.t.1.7 | $84$ | $2$ | $2$ | $2$ |
| 140.96.2-140.b.1.1 | $140$ | $2$ | $2$ | $2$ |
| 140.96.2-140.j.1.1 | $140$ | $2$ | $2$ | $2$ |
| 154.96.2-154.c.1.3 | $154$ | $2$ | $2$ | $2$ |
| 154.96.2-154.d.1.3 | $154$ | $2$ | $2$ | $2$ |
| 168.96.2-168.n.1.13 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.o.1.13 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.bl.1.13 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.bm.1.13 | $168$ | $2$ | $2$ | $2$ |
| 182.96.2-182.l.1.3 | $182$ | $2$ | $2$ | $2$ |
| 182.96.2-182.m.1.3 | $182$ | $2$ | $2$ | $2$ |
| 210.96.2-210.b.1.6 | $210$ | $2$ | $2$ | $2$ |
| 210.96.2-210.f.1.6 | $210$ | $2$ | $2$ | $2$ |
| 238.96.2-238.c.1.3 | $238$ | $2$ | $2$ | $2$ |
| 238.96.2-238.d.1.3 | $238$ | $2$ | $2$ | $2$ |
| 266.96.2-266.l.1.4 | $266$ | $2$ | $2$ | $2$ |
| 266.96.2-266.m.1.4 | $266$ | $2$ | $2$ | $2$ |
| 280.96.2-280.f.1.1 | $280$ | $2$ | $2$ | $2$ |
| 280.96.2-280.g.1.1 | $280$ | $2$ | $2$ | $2$ |
| 280.96.2-280.r.1.1 | $280$ | $2$ | $2$ | $2$ |
| 280.96.2-280.s.1.1 | $280$ | $2$ | $2$ | $2$ |
| 308.96.2-308.c.1.3 | $308$ | $2$ | $2$ | $2$ |
| 308.96.2-308.d.1.3 | $308$ | $2$ | $2$ | $2$ |
| 322.96.2-322.c.1.3 | $322$ | $2$ | $2$ | $2$ |
| 322.96.2-322.d.1.3 | $322$ | $2$ | $2$ | $2$ |