Invariants
Level: | $56$ | $\SL_2$-level: | $4$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
Elliptic points: | $0$ 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: | 4E0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.0.89 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}3&48\\2&33\end{bmatrix}$, $\begin{bmatrix}7&2\\24&33\end{bmatrix}$, $\begin{bmatrix}23&10\\20&7\end{bmatrix}$, $\begin{bmatrix}31&36\\28&51\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.12.0.a.1 for the level structure with $-I$) |
Cyclic 56-isogeny field degree: | $32$ |
Cyclic 56-torsion field degree: | $768$ |
Full 56-torsion field degree: | $129024$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 298 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3^4\cdot7^2}\cdot\frac{x^{12}(196x^{4}+126x^{2}y^{2}+81y^{4})^{3}}{y^{4}x^{16}(14x^{2}+9y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.12.0-2.a.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
28.12.0-2.a.1.1 | $28$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
56.48.0-56.a.1.2 | $56$ | $2$ | $2$ | $0$ |
56.48.0-56.a.1.3 | $56$ | $2$ | $2$ | $0$ |
56.48.0-56.b.1.1 | $56$ | $2$ | $2$ | $0$ |
56.48.0-56.b.1.6 | $56$ | $2$ | $2$ | $0$ |
56.48.0-56.e.1.3 | $56$ | $2$ | $2$ | $0$ |
56.48.0-56.e.1.5 | $56$ | $2$ | $2$ | $0$ |
56.48.0-56.g.1.3 | $56$ | $2$ | $2$ | $0$ |
56.48.0-56.g.1.8 | $56$ | $2$ | $2$ | $0$ |
56.192.5-56.c.1.3 | $56$ | $8$ | $8$ | $5$ |
56.504.16-56.c.1.4 | $56$ | $21$ | $21$ | $16$ |
56.672.21-56.c.1.10 | $56$ | $28$ | $28$ | $21$ |
168.48.0-168.h.1.3 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.h.1.11 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.j.1.5 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.j.1.13 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.n.1.9 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.n.1.10 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.p.1.3 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.p.1.4 | $168$ | $2$ | $2$ | $0$ |
168.72.2-168.a.1.8 | $168$ | $3$ | $3$ | $2$ |
168.96.1-168.dg.1.9 | $168$ | $4$ | $4$ | $1$ |
280.48.0-280.h.1.9 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.h.1.12 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.j.1.5 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.j.1.8 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.n.1.6 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.n.1.10 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.p.1.7 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.p.1.11 | $280$ | $2$ | $2$ | $0$ |
280.120.4-280.a.1.4 | $280$ | $5$ | $5$ | $4$ |
280.144.3-280.a.1.1 | $280$ | $6$ | $6$ | $3$ |
280.240.7-280.a.1.3 | $280$ | $10$ | $10$ | $7$ |