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: | $6$ (of which $3$ are rational) | Cusp widths | $1^{3}\cdot7^{3}$ | Cusp orbits | $1^{3}\cdot3$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 7E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 7.48.0.5 |
| Sutherland (S) label: | 7B.1.3 |
Level structure
| $\GL_2(\Z/7\Z)$-generators: | $\begin{bmatrix}4&3\\0&1\end{bmatrix}$, $\begin{bmatrix}6&3\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/7\Z)$-subgroup: | $F_7$ |
| Contains $-I$: | no $\quad$ (see 7.24.0.a.2 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 80 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{7^4}{2}\cdot\frac{(x-4y)^{24}(12x^{2}-30xy+49y^{2})^{3}(2752x^{6}-45984x^{5}y+220480x^{4}y^{2}-278440x^{3}y^{3}-134000x^{2}y^{4}+267574xy^{5}+37969y^{6})^{3}}{(x-4y)^{24}(2x-19y)(5x-9y)(8x+y)(8x^{3}-228x^{2}y+472xy^{2}-83y^{3})^{7}}$ |
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 |
|---|---|---|---|---|
| $X_{\mathrm{arith}}(7)$ | $7$ | $7$ | $7$ | $3$ |
| 14.96.2-14.c.2.2 | $14$ | $2$ | $2$ | $2$ |
| 14.96.2-14.e.2.2 | $14$ | $2$ | $2$ | $2$ |
| 14.144.1-14.a.2.1 | $14$ | $3$ | $3$ | $1$ |
| 21.144.4-21.a.2.2 | $21$ | $3$ | $3$ | $4$ |
| 21.192.3-21.a.2.2 | $21$ | $4$ | $4$ | $3$ |
| 28.96.2-28.d.2.6 | $28$ | $2$ | $2$ | $2$ |
| 28.96.2-28.f.2.4 | $28$ | $2$ | $2$ | $2$ |
| 28.192.6-28.k.1.3 | $28$ | $4$ | $4$ | $6$ |
| 35.240.8-35.a.1.1 | $35$ | $5$ | $5$ | $8$ |
| 35.288.7-35.a.1.7 | $35$ | $6$ | $6$ | $7$ |
| 35.480.15-35.a.1.2 | $35$ | $10$ | $10$ | $15$ |
| 42.96.2-42.a.2.4 | $42$ | $2$ | $2$ | $2$ |
| 42.96.2-42.e.2.4 | $42$ | $2$ | $2$ | $2$ |
| 49.336.3-49.b.2.1 | $49$ | $7$ | $7$ | $3$ |
| 56.96.2-56.f.2.7 | $56$ | $2$ | $2$ | $2$ |
| 56.96.2-56.g.2.7 | $56$ | $2$ | $2$ | $2$ |
| 56.96.2-56.j.2.7 | $56$ | $2$ | $2$ | $2$ |
| 56.96.2-56.k.2.7 | $56$ | $2$ | $2$ | $2$ |
| 63.1296.46-63.a.1.3 | $63$ | $27$ | $27$ | $46$ |
| 70.96.2-70.a.2.3 | $70$ | $2$ | $2$ | $2$ |
| 70.96.2-70.d.2.3 | $70$ | $2$ | $2$ | $2$ |
| 84.96.2-84.g.2.6 | $84$ | $2$ | $2$ | $2$ |
| 84.96.2-84.p.2.6 | $84$ | $2$ | $2$ | $2$ |
| 140.96.2-140.a.2.5 | $140$ | $2$ | $2$ | $2$ |
| 140.96.2-140.f.2.5 | $140$ | $2$ | $2$ | $2$ |
| 154.96.2-154.a.2.3 | $154$ | $2$ | $2$ | $2$ |
| 154.96.2-154.b.2.3 | $154$ | $2$ | $2$ | $2$ |
| 168.96.2-168.l.2.10 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.m.2.10 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.bf.2.10 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.bg.2.10 | $168$ | $2$ | $2$ | $2$ |
| 182.96.2-182.j.2.3 | $182$ | $2$ | $2$ | $2$ |
| 182.96.2-182.k.2.3 | $182$ | $2$ | $2$ | $2$ |
| 210.96.2-210.a.2.2 | $210$ | $2$ | $2$ | $2$ |
| 210.96.2-210.d.2.2 | $210$ | $2$ | $2$ | $2$ |
| 238.96.2-238.a.2.3 | $238$ | $2$ | $2$ | $2$ |
| 238.96.2-238.b.2.3 | $238$ | $2$ | $2$ | $2$ |
| 266.96.2-266.j.2.4 | $266$ | $2$ | $2$ | $2$ |
| 266.96.2-266.k.2.4 | $266$ | $2$ | $2$ | $2$ |
| 280.96.2-280.d.2.9 | $280$ | $2$ | $2$ | $2$ |
| 280.96.2-280.e.2.9 | $280$ | $2$ | $2$ | $2$ |
| 280.96.2-280.l.2.9 | $280$ | $2$ | $2$ | $2$ |
| 280.96.2-280.m.2.9 | $280$ | $2$ | $2$ | $2$ |
| 308.96.2-308.a.2.5 | $308$ | $2$ | $2$ | $2$ |
| 308.96.2-308.b.2.5 | $308$ | $2$ | $2$ | $2$ |
| 322.96.2-322.a.2.3 | $322$ | $2$ | $2$ | $2$ |
| 322.96.2-322.b.2.3 | $322$ | $2$ | $2$ | $2$ |