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 $\quad(D =$ $-7$) |
Other labels
| Cummins and Pauli (CP) label: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.1.57 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&42\\30&15\end{bmatrix}$, $\begin{bmatrix}7&20\\36&35\end{bmatrix}$, $\begin{bmatrix}14&19\\19&22\end{bmatrix}$, $\begin{bmatrix}38&21\\31&2\end{bmatrix}$ |
| 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 2^8\,\frac{795674880x^{2}y^{14}+51348120768x^{2}y^{13}z-53088318960817x^{2}y^{12}z^{2}-13584812515590456x^{2}y^{11}z^{3}-1580811031325295224x^{2}y^{10}z^{4}-68756634794459423616x^{2}y^{9}z^{5}-1145547393642261082893x^{2}y^{8}z^{6}+6529088638038906236304x^{2}y^{7}z^{7}+645082180256258726906792x^{2}y^{6}z^{8}+13462293530736216873495936x^{2}y^{5}z^{9}+158304769056859380257709855x^{2}y^{4}z^{10}+1169837631772025759683147728x^{2}y^{3}z^{11}+5406552435575262922363838880x^{2}y^{2}z^{12}+14305805542705842224907321600x^{2}yz^{13}+16599911586684598027687104000x^{2}z^{14}+43545600xy^{15}+6286910112xy^{14}z-595011709488xy^{13}z^{2}+329195716681720xy^{12}z^{3}+73682991670243872xy^{11}z^{4}-4480810386224433380xy^{10}z^{5}-707307349847824652136xy^{9}z^{6}-30987768523100013427264xy^{8}z^{7}-708629396923130523658272xy^{7}z^{8}-9732074710381817419766637xy^{6}z^{9}-83577073608104302254706776xy^{5}z^{10}-441219006190418629911454080xy^{4}z^{11}-1314188474567109818865552000xy^{3}z^{12}-1693454886145043590375296000xy^{2}z^{13}+864000y^{16}+5649654528y^{15}z+1646420322064y^{14}z^{2}+275719542238560y^{13}z^{3}+45908850626242737y^{12}z^{4}+6168408422786713320y^{11}z^{5}+327053205980223672208y^{10}z^{6}+8694484612047831468480y^{9}z^{7}+132207674634295069627181y^{8}z^{8}+1218124114217012746019640y^{7}z^{9}+6778704026143869038416624y^{6}z^{10}+20816947737700022790399168y^{5}z^{11}+27866742741922614412593312y^{4}z^{12}-9199974256891539531363072y^{3}z^{13}+63488701911280473182234880y^{2}z^{14}-170255503453175364386534400yz^{15}+165526183912809382042464000z^{16}}{28442624x^{2}y^{14}-6237355755008x^{2}y^{12}z^{2}-6249101022818528x^{2}y^{10}z^{4}-479016410780014863x^{2}y^{8}z^{6}+66542671646204336576x^{2}y^{6}z^{8}-2144688503626423319040x^{2}y^{4}z^{10}+22597899344775582564352x^{2}y^{2}z^{12}-4539924480xy^{14}z-84271109038592xy^{12}z^{3}-21946042885027009xy^{10}z^{5}+2683581791795795072xy^{8}z^{7}-121512919889396593152xy^{6}z^{9}+2881432265921798012928xy^{4}z^{11}-31375595784481800585216xy^{2}z^{13}-65536y^{16}+307311140864y^{14}z^{2}-1388210726275520y^{12}z^{4}-139045350556782656y^{10}z^{6}-39012420499121043968y^{8}z^{8}+4301107813958502916096y^{6}z^{10}-151088434462177242103808y^{4}z^{12}+2269501428410850242330624y^{2}z^{14}-12555467579756800534183936z^{16}}$ |
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.z.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 28.12.0.k.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.ho.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.hp.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.hq.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.hr.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.jc.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.jd.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.je.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1.jf.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.192.13.fa.1 | $56$ | $8$ | $8$ | $13$ | $8$ | $1^{8}\cdot2^{2}$ |
| 56.504.37.oo.1 | $56$ | $21$ | $21$ | $37$ | $19$ | $1^{4}\cdot2^{14}\cdot4$ |
| 56.672.49.oo.1 | $56$ | $28$ | $28$ | $49$ | $26$ | $1^{12}\cdot2^{16}\cdot4$ |
| 112.48.3.bl.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.bl.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.dv.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.dv.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.dx.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.dx.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.gd.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.gd.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.1.ckq.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ckr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.cks.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ckt.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.clw.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.clx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.cly.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.clz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.5.bru.1 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 168.96.5.uq.1 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 280.48.1.cew.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.cex.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.cey.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.cez.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.cgc.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.cgd.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.cge.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.cgf.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.120.9.og.1 | $280$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 280.144.9.bne.1 | $280$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 280.240.17.fhs.1 | $280$ | $10$ | $10$ | $17$ | $?$ | not computed |