Invariants
| Level: | $7$ | $\SL_2$-level: | $7$ | ||||
| Index: | $28$ | $\PSL_2$-index: | $28$ | ||||
| Genus: | $0 = 1 + \frac{ 28 }{12} - \frac{ 4 }{4} - \frac{ 1 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $1$ is rational) | Cusp widths | $7^{4}$ | Cusp orbits | $1\cdot3$ | ||
| Elliptic points: | $4$ of order $2$ and $1$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-12,-19,-27$) |
Other labels
| Cummins and Pauli (CP) label: | 7F0 |
| Sutherland and Zywina (SZ) label: | 7F0-7a |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 7.28.0.1 |
| Sutherland (S) label: | 7Ns |
Level structure
| $\GL_2(\Z/7\Z)$-generators: | $\begin{bmatrix}0&3\\4&0\end{bmatrix}$, $\begin{bmatrix}1&0\\0&5\end{bmatrix}$ |
| $\GL_2(\Z/7\Z)$-subgroup: | $C_6\wr C_2$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 7-isogeny field degree: | $2$ |
| Cyclic 7-torsion field degree: | $12$ |
| Full 7-torsion field degree: | $72$ |
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 28 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^7}\cdot\frac{(x-y)^{29}(x+y)^{3}(x^{2}-12xy+15y^{2})^{3}(x^{2}-12xy+43y^{2})^{3}(x^{4}-14x^{3}y+68x^{2}y^{2}-154xy^{3}+211y^{4})^{3}}{y^{7}(x-y)^{28}(x^{3}-11x^{2}y+31xy^{2}-13y^{3})^{7}}$ |
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 |
|---|---|---|---|---|---|
| $X(1)$ | $1$ | $28$ | $28$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| $X_{\mathrm{sp}}(7)$ | $7$ | $2$ | $2$ | $1$ |
| 7.56.1.b.1 | $7$ | $2$ | $2$ | $1$ |
| 7.84.1.a.1 | $7$ | $3$ | $3$ | $1$ |
| 14.56.1.a.1 | $14$ | $2$ | $2$ | $1$ |
| 14.56.1.b.1 | $14$ | $2$ | $2$ | $1$ |
| 14.56.3.a.1 | $14$ | $2$ | $2$ | $3$ |
| 14.56.3.b.1 | $14$ | $2$ | $2$ | $3$ |
| 14.84.3.a.1 | $14$ | $3$ | $3$ | $3$ |
| 21.56.1.a.1 | $21$ | $2$ | $2$ | $1$ |
| 21.56.1.b.1 | $21$ | $2$ | $2$ | $1$ |
| 21.84.3.a.1 | $21$ | $3$ | $3$ | $3$ |
| 21.112.6.a.1 | $21$ | $4$ | $4$ | $6$ |
| 28.56.1.a.1 | $28$ | $2$ | $2$ | $1$ |
| 28.56.1.b.1 | $28$ | $2$ | $2$ | $1$ |
| 28.56.1.c.1 | $28$ | $2$ | $2$ | $1$ |
| 28.56.1.d.1 | $28$ | $2$ | $2$ | $1$ |
| 28.56.3.a.1 | $28$ | $2$ | $2$ | $3$ |
| 28.56.3.b.1 | $28$ | $2$ | $2$ | $3$ |
| 28.112.6.a.1 | $28$ | $4$ | $4$ | $6$ |
| 35.56.1.a.1 | $35$ | $2$ | $2$ | $1$ |
| 35.56.1.b.1 | $35$ | $2$ | $2$ | $1$ |
| 35.140.9.a.1 | $35$ | $5$ | $5$ | $9$ |
| 35.168.9.a.1 | $35$ | $6$ | $6$ | $9$ |
| 35.280.18.a.1 | $35$ | $10$ | $10$ | $18$ |
| 42.56.1.a.1 | $42$ | $2$ | $2$ | $1$ |
| 42.56.1.b.1 | $42$ | $2$ | $2$ | $1$ |
| 42.56.3.a.1 | $42$ | $2$ | $2$ | $3$ |
| 42.56.3.b.1 | $42$ | $2$ | $2$ | $3$ |
| 49.196.9.a.1 | $49$ | $7$ | $7$ | $9$ |
| $X_{\mathrm{sp}}^+(49)$ | $49$ | $49$ | $49$ | $94$ |
| 56.56.1.a.1 | $56$ | $2$ | $2$ | $1$ |
| 56.56.1.b.1 | $56$ | $2$ | $2$ | $1$ |
| 56.56.1.c.1 | $56$ | $2$ | $2$ | $1$ |
| 56.56.1.d.1 | $56$ | $2$ | $2$ | $1$ |
| 56.56.1.e.1 | $56$ | $2$ | $2$ | $1$ |
| 56.56.1.f.1 | $56$ | $2$ | $2$ | $1$ |
| 56.56.1.g.1 | $56$ | $2$ | $2$ | $1$ |
| 56.56.1.h.1 | $56$ | $2$ | $2$ | $1$ |
| 56.56.3.a.1 | $56$ | $2$ | $2$ | $3$ |
| 56.56.3.b.1 | $56$ | $2$ | $2$ | $3$ |
| 56.56.3.c.1 | $56$ | $2$ | $2$ | $3$ |
| 56.56.3.d.1 | $56$ | $2$ | $2$ | $3$ |
| 63.756.54.a.1 | $63$ | $27$ | $27$ | $54$ |
| 70.56.1.a.1 | $70$ | $2$ | $2$ | $1$ |
| 70.56.1.b.1 | $70$ | $2$ | $2$ | $1$ |
| 70.56.3.a.1 | $70$ | $2$ | $2$ | $3$ |
| 70.56.3.b.1 | $70$ | $2$ | $2$ | $3$ |
| 77.56.1.a.1 | $77$ | $2$ | $2$ | $1$ |
| 77.56.1.b.1 | $77$ | $2$ | $2$ | $1$ |
| 84.56.1.a.1 | $84$ | $2$ | $2$ | $1$ |
| 84.56.1.b.1 | $84$ | $2$ | $2$ | $1$ |
| 84.56.1.c.1 | $84$ | $2$ | $2$ | $1$ |
| 84.56.1.d.1 | $84$ | $2$ | $2$ | $1$ |
| 84.56.3.a.1 | $84$ | $2$ | $2$ | $3$ |
| 84.56.3.b.1 | $84$ | $2$ | $2$ | $3$ |
| 91.56.1.a.1 | $91$ | $2$ | $2$ | $1$ |
| 91.56.1.b.1 | $91$ | $2$ | $2$ | $1$ |
| 105.56.1.a.1 | $105$ | $2$ | $2$ | $1$ |
| 105.56.1.b.1 | $105$ | $2$ | $2$ | $1$ |
| 119.56.1.a.1 | $119$ | $2$ | $2$ | $1$ |
| 119.56.1.b.1 | $119$ | $2$ | $2$ | $1$ |
| 133.56.1.a.1 | $133$ | $2$ | $2$ | $1$ |
| 133.56.1.b.1 | $133$ | $2$ | $2$ | $1$ |
| 140.56.1.a.1 | $140$ | $2$ | $2$ | $1$ |
| 140.56.1.b.1 | $140$ | $2$ | $2$ | $1$ |
| 140.56.1.c.1 | $140$ | $2$ | $2$ | $1$ |
| 140.56.1.d.1 | $140$ | $2$ | $2$ | $1$ |
| 140.56.3.a.1 | $140$ | $2$ | $2$ | $3$ |
| 140.56.3.b.1 | $140$ | $2$ | $2$ | $3$ |
| 154.56.1.a.1 | $154$ | $2$ | $2$ | $1$ |
| 154.56.1.b.1 | $154$ | $2$ | $2$ | $1$ |
| 154.56.3.a.1 | $154$ | $2$ | $2$ | $3$ |
| 154.56.3.b.1 | $154$ | $2$ | $2$ | $3$ |
| 161.56.1.a.1 | $161$ | $2$ | $2$ | $1$ |
| 161.56.1.b.1 | $161$ | $2$ | $2$ | $1$ |
| 168.56.1.a.1 | $168$ | $2$ | $2$ | $1$ |
| 168.56.1.b.1 | $168$ | $2$ | $2$ | $1$ |
| 168.56.1.c.1 | $168$ | $2$ | $2$ | $1$ |
| 168.56.1.d.1 | $168$ | $2$ | $2$ | $1$ |
| 168.56.1.e.1 | $168$ | $2$ | $2$ | $1$ |
| 168.56.1.f.1 | $168$ | $2$ | $2$ | $1$ |
| 168.56.1.g.1 | $168$ | $2$ | $2$ | $1$ |
| 168.56.1.h.1 | $168$ | $2$ | $2$ | $1$ |
| 168.56.3.a.1 | $168$ | $2$ | $2$ | $3$ |
| 168.56.3.b.1 | $168$ | $2$ | $2$ | $3$ |
| 168.56.3.c.1 | $168$ | $2$ | $2$ | $3$ |
| 168.56.3.d.1 | $168$ | $2$ | $2$ | $3$ |
| 182.56.1.g.1 | $182$ | $2$ | $2$ | $1$ |
| 182.56.1.h.1 | $182$ | $2$ | $2$ | $1$ |
| 182.56.3.a.1 | $182$ | $2$ | $2$ | $3$ |
| 182.56.3.b.1 | $182$ | $2$ | $2$ | $3$ |
| 203.56.1.a.1 | $203$ | $2$ | $2$ | $1$ |
| 203.56.1.b.1 | $203$ | $2$ | $2$ | $1$ |
| 210.56.1.a.1 | $210$ | $2$ | $2$ | $1$ |
| 210.56.1.b.1 | $210$ | $2$ | $2$ | $1$ |
| 210.56.3.a.1 | $210$ | $2$ | $2$ | $3$ |
| 210.56.3.b.1 | $210$ | $2$ | $2$ | $3$ |
| 217.56.1.a.1 | $217$ | $2$ | $2$ | $1$ |
| 217.56.1.b.1 | $217$ | $2$ | $2$ | $1$ |
| 231.56.1.a.1 | $231$ | $2$ | $2$ | $1$ |
| 231.56.1.b.1 | $231$ | $2$ | $2$ | $1$ |
| 238.56.1.a.1 | $238$ | $2$ | $2$ | $1$ |
| 238.56.1.b.1 | $238$ | $2$ | $2$ | $1$ |
| 238.56.3.a.1 | $238$ | $2$ | $2$ | $3$ |
| 238.56.3.b.1 | $238$ | $2$ | $2$ | $3$ |
| 259.56.1.a.1 | $259$ | $2$ | $2$ | $1$ |
| 259.56.1.b.1 | $259$ | $2$ | $2$ | $1$ |
| 266.56.1.a.1 | $266$ | $2$ | $2$ | $1$ |
| 266.56.1.b.1 | $266$ | $2$ | $2$ | $1$ |
| 266.56.3.a.1 | $266$ | $2$ | $2$ | $3$ |
| 266.56.3.b.1 | $266$ | $2$ | $2$ | $3$ |
| 273.56.1.a.1 | $273$ | $2$ | $2$ | $1$ |
| 273.56.1.b.1 | $273$ | $2$ | $2$ | $1$ |
| 280.56.1.a.1 | $280$ | $2$ | $2$ | $1$ |
| 280.56.1.b.1 | $280$ | $2$ | $2$ | $1$ |
| 280.56.1.c.1 | $280$ | $2$ | $2$ | $1$ |
| 280.56.1.d.1 | $280$ | $2$ | $2$ | $1$ |
| 280.56.1.e.1 | $280$ | $2$ | $2$ | $1$ |
| 280.56.1.f.1 | $280$ | $2$ | $2$ | $1$ |
| 280.56.1.g.1 | $280$ | $2$ | $2$ | $1$ |
| 280.56.1.h.1 | $280$ | $2$ | $2$ | $1$ |
| 280.56.3.a.1 | $280$ | $2$ | $2$ | $3$ |
| 280.56.3.b.1 | $280$ | $2$ | $2$ | $3$ |
| 280.56.3.c.1 | $280$ | $2$ | $2$ | $3$ |
| 280.56.3.d.1 | $280$ | $2$ | $2$ | $3$ |
| 287.56.1.a.1 | $287$ | $2$ | $2$ | $1$ |
| 287.56.1.b.1 | $287$ | $2$ | $2$ | $1$ |
| 301.56.1.a.1 | $301$ | $2$ | $2$ | $1$ |
| 301.56.1.b.1 | $301$ | $2$ | $2$ | $1$ |
| 308.56.1.a.1 | $308$ | $2$ | $2$ | $1$ |
| 308.56.1.b.1 | $308$ | $2$ | $2$ | $1$ |
| 308.56.1.c.1 | $308$ | $2$ | $2$ | $1$ |
| 308.56.1.d.1 | $308$ | $2$ | $2$ | $1$ |
| 308.56.3.a.1 | $308$ | $2$ | $2$ | $3$ |
| 308.56.3.b.1 | $308$ | $2$ | $2$ | $3$ |
| 322.56.1.a.1 | $322$ | $2$ | $2$ | $1$ |
| 322.56.1.b.1 | $322$ | $2$ | $2$ | $1$ |
| 322.56.3.a.1 | $322$ | $2$ | $2$ | $3$ |
| 322.56.3.b.1 | $322$ | $2$ | $2$ | $3$ |
| 329.56.1.a.1 | $329$ | $2$ | $2$ | $1$ |
| 329.56.1.b.1 | $329$ | $2$ | $2$ | $1$ |