Invariants
Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8B1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.1.4 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}13&34\\46&15\end{bmatrix}$, $\begin{bmatrix}31&8\\52&39\end{bmatrix}$, $\begin{bmatrix}35&10\\4&33\end{bmatrix}$, $\begin{bmatrix}43&12\\42&11\end{bmatrix}$, $\begin{bmatrix}53&20\\20&5\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 56.48.1-56.b.1.1, 56.48.1-56.b.1.2, 56.48.1-56.b.1.3, 56.48.1-56.b.1.4, 56.48.1-56.b.1.5, 56.48.1-56.b.1.6, 56.48.1-56.b.1.7, 56.48.1-56.b.1.8, 168.48.1-56.b.1.1, 168.48.1-56.b.1.2, 168.48.1-56.b.1.3, 168.48.1-56.b.1.4, 168.48.1-56.b.1.5, 168.48.1-56.b.1.6, 168.48.1-56.b.1.7, 168.48.1-56.b.1.8, 280.48.1-56.b.1.1, 280.48.1-56.b.1.2, 280.48.1-56.b.1.3, 280.48.1-56.b.1.4, 280.48.1-56.b.1.5, 280.48.1-56.b.1.6, 280.48.1-56.b.1.7, 280.48.1-56.b.1.8 |
Cyclic 56-isogeny field degree: | $32$ |
Cyclic 56-torsion field degree: | $768$ |
Full 56-torsion field degree: | $129024$ |
Jacobian
Conductor: | $2^{6}\cdot7^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 3136.2.a.m |
Models
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} + 49x $ |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{7^4}\cdot\frac{7203x^{2}y^{4}z^{2}-49xy^{6}z+17294403xy^{2}z^{5}+y^{8}+13841287201z^{8}}{z^{2}y^{4}x^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
4.12.0.a.1 | $4$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.12.0.bx.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.12.1.c.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
56.48.1.a.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.d.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.n.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.z.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.bk.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.bn.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.bp.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.br.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.192.13.f.1 | $56$ | $8$ | $8$ | $13$ | $5$ | $1^{8}\cdot2^{2}$ |
56.504.37.f.1 | $56$ | $21$ | $21$ | $37$ | $13$ | $1^{4}\cdot2^{14}\cdot4$ |
56.672.49.f.1 | $56$ | $28$ | $28$ | $49$ | $17$ | $1^{12}\cdot2^{16}\cdot4$ |
168.48.1.bd.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.bf.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.bp.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.br.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.en.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.ep.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.er.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.et.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.72.5.b.1 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
168.96.5.b.1 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
280.48.1.bd.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bf.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bp.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.br.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.ej.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.el.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.en.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.ep.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.120.9.b.1 | $280$ | $5$ | $5$ | $9$ | $?$ | not computed |
280.144.9.b.1 | $280$ | $6$ | $6$ | $9$ | $?$ | not computed |
280.240.17.fp.1 | $280$ | $10$ | $10$ | $17$ | $?$ | not computed |