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: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | yes $\quad(D =$ $-4,-7$) |
Other labels
Cummins and Pauli (CP) label: | 8C1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.1.46 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}18&13\\19&38\end{bmatrix}$, $\begin{bmatrix}36&13\\27&24\end{bmatrix}$, $\begin{bmatrix}39&42\\22&13\end{bmatrix}$, $\begin{bmatrix}52&23\\29&8\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
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} - 2156x - 38416 $ |
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{10584x^{2}y^{6}+135646896x^{2}y^{4}z^{2}+1057095894500352x^{2}y^{2}z^{4}+90157548823944254464x^{2}z^{6}-1118376xy^{6}z+142088798208xy^{4}z^{3}+70600409898366208xy^{2}z^{5}+4832263270955810213888xz^{7}-27y^{8}-5751424y^{6}z^{2}+14620551823360y^{4}z^{4}+2252972524307642368y^{2}z^{6}+64619851778054756356096z^{8}}{280x^{2}y^{6}+161616112x^{2}y^{4}z^{2}+843308032x^{2}y^{2}z^{4}+289254654976x^{2}z^{6}+37240xy^{6}z+8699841024xy^{4}z^{3}-22136835840xy^{2}z^{5}-8099130339328xz^{7}+y^{8}+2765952y^{6}z^{2}+117099343872y^{4}z^{4}-1239662807040y^{2}z^{6}-396857386627072z^{8}}$ |
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 |
---|---|---|---|---|---|---|
8.12.0.v.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
28.12.0.n.1 | $28$ | $2$ | $2$ | $0$ | $0$ | 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.bp.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.ci.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.eo.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.er.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.jb.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.jf.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.jq.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.ju.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.192.13.el.1 | $56$ | $8$ | $8$ | $13$ | $7$ | $1^{12}$ |
56.504.37.md.1 | $56$ | $21$ | $21$ | $37$ | $20$ | $1^{8}\cdot2^{12}\cdot4$ |
56.672.49.md.1 | $56$ | $28$ | $28$ | $49$ | $26$ | $1^{20}\cdot2^{12}\cdot4$ |
112.48.2.cn.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
112.48.2.cp.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
112.48.2.dl.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
112.48.2.dn.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
112.48.2.ej.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
112.48.2.el.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
112.48.2.er.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
112.48.2.et.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
168.48.1.bdn.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.bdv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.bfj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.bfr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.cih.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.cip.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.cjm.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.cju.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.72.5.bdr.1 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
168.96.5.qj.1 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
280.48.1.ban.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bav.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bcj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.bcr.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.ccn.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.ccv.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.cds.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.cea.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.120.9.jr.1 | $280$ | $5$ | $5$ | $9$ | $?$ | not computed |
280.144.9.zb.1 | $280$ | $6$ | $6$ | $9$ | $?$ | not computed |
280.240.17.dov.1 | $280$ | $10$ | $10$ | $17$ | $?$ | not computed |