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 $\quad(D =$ $-4$) |
Other labels
Cummins and Pauli (CP) label: | 8B1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.1.43 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}2&49\\15&18\end{bmatrix}$, $\begin{bmatrix}20&53\\47&0\end{bmatrix}$, $\begin{bmatrix}22&1\\3&50\end{bmatrix}$, $\begin{bmatrix}33&28\\28&43\end{bmatrix}$ |
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} + 49x $ |
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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{3780x^{2}y^{6}-47302101x^{2}y^{4}z^{2}-6670698300x^{2}y^{2}z^{4}-495744061995x^{2}z^{6}-194922xy^{6}z+154288260xy^{4}z^{3}+19767502629xy^{2}z^{5}-27y^{8}+4596200y^{6}z^{2}-677658240y^{4}z^{4}+21790947780y^{2}z^{6}-373714754427z^{8}}{28x^{2}y^{6}-40817x^{2}y^{4}z^{2}-49412580x^{2}y^{2}z^{4}+18360891185x^{2}z^{6}-98xy^{6}z+1142876xy^{4}z^{3}-732129727xy^{2}z^{5}+y^{8}-8232y^{6}z^{2}+7529536y^{4}z^{4}+161414428y^{2}z^{6}+13841287201z^{8}}$ |
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.y.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.12.0.bn.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.ia.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.ib.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.ic.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.id.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.jo.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.jp.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.jq.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.48.1.jr.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
56.192.13.fd.1 | $56$ | $8$ | $8$ | $13$ | $6$ | $1^{8}\cdot2^{2}$ |
56.504.37.or.1 | $56$ | $21$ | $21$ | $37$ | $18$ | $1^{4}\cdot2^{14}\cdot4$ |
56.672.49.or.1 | $56$ | $28$ | $28$ | $49$ | $23$ | $1^{12}\cdot2^{16}\cdot4$ |
168.48.1.clc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.cld.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.cle.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.clf.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.cmi.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.cmj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.cmk.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.cml.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.72.5.brx.1 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
168.96.5.ut.1 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
280.48.1.cfi.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.cfj.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.cfk.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.cfl.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.cgo.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.cgp.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.cgq.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.48.1.cgr.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
280.120.9.oj.1 | $280$ | $5$ | $5$ | $9$ | $?$ | not computed |
280.144.9.bnh.1 | $280$ | $6$ | $6$ | $9$ | $?$ | not computed |
280.240.17.fhv.1 | $280$ | $10$ | $10$ | $17$ | $?$ | not computed |