Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot2\cdot7^{2}\cdot8\cdot14\cdot56$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 56D5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.192.5.433 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}6&41\\11&24\end{bmatrix}$, $\begin{bmatrix}15&6\\36&21\end{bmatrix}$, $\begin{bmatrix}18&33\\21&22\end{bmatrix}$, $\begin{bmatrix}40&21\\15&30\end{bmatrix}$, $\begin{bmatrix}53&8\\24&49\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.96.5.bp.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $2$ |
| Cyclic 56-torsion field degree: | $48$ |
| Full 56-torsion field degree: | $16128$ |
Jacobian
| Conductor: | $2^{20}\cdot7^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}$ |
| Newforms: | 14.2.a.a$^{2}$, 3136.2.a.e, 3136.2.a.q, 3136.2.a.w |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ - x w + x t + y z $ |
| $=$ | $14 x^{2} + 14 x y - z^{2} + z w + 2 w^{2} - w t$ | |
| $=$ | $28 x^{2} - 14 x y + 14 y^{2} + z^{2} + z w - z t$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{7} + 6 x^{6} y + 13 x^{5} y^{2} + 112 x^{5} z^{2} + 4 x^{4} y^{3} + 182 x^{4} y z^{2} + \cdots + 14 y^{5} z^{2} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:1/2:1)$, $(0:0:-1:1:0)$, $(0:0:1/2:1/2:1)$, $(0:0:0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{210830454904832y^{2}w^{10}-1060006164417536y^{2}w^{9}t+1894259019695616y^{2}w^{8}t^{2}-1332482329941504y^{2}w^{7}t^{3}-170831007881856y^{2}w^{6}t^{4}+967563955157184y^{2}w^{5}t^{5}-765536598491616y^{2}w^{4}t^{6}+332316923981280y^{2}w^{3}t^{7}-92778838141752y^{2}w^{2}t^{8}+16532438140148y^{2}wt^{9}-1462840306046y^{2}t^{10}+42038421000192z^{2}w^{10}-75408182633472z^{2}w^{9}t-52108183636992z^{2}w^{8}t^{2}+272982001363200z^{2}w^{7}t^{3}-352145262216384z^{2}w^{6}t^{4}+247587200159616z^{2}w^{5}t^{5}-106144908213312z^{2}w^{4}t^{6}+27682290703296z^{2}w^{3}t^{7}-4274445553668z^{2}w^{2}t^{8}+479996341884z^{2}wt^{9}-55064600610z^{2}t^{10}+32427907094528zw^{11}-136684103625216zw^{10}t+252809685611776zw^{9}t^{2}-236921311364352zw^{8}t^{3}+69477771446400zw^{7}t^{4}+87461064775008zw^{6}t^{5}-119555870735952zw^{5}t^{6}+73185698464800zw^{4}t^{7}-28095053838168zw^{3}t^{8}+7328288798654zw^{2}t^{9}-1246342932483zwt^{10}+104448282505zt^{11}-8820093334528w^{12}-45501825844224w^{11}t+292076762094592w^{10}t^{2}-593976370873344w^{9}t^{3}+627996125702784w^{8}t^{4}-382042256491584w^{7}t^{5}+126182437858944w^{6}t^{6}-9492914498016w^{5}t^{7}-10497425909880w^{4}t^{8}+4946569652588w^{3}t^{9}-978200913840w^{2}t^{10}+76805127544wt^{11}+6718464t^{12}}{300472008704y^{2}w^{10}+3604569751552y^{2}w^{9}t-23433370619904y^{2}w^{8}t^{2}+60065466553344y^{2}w^{7}t^{3}-87401538491904y^{2}w^{6}t^{4}+79838740832256y^{2}w^{5}t^{5}-47388561921792y^{2}w^{4}t^{6}+18275117226576y^{2}w^{3}t^{7}-4421761140384y^{2}w^{2}t^{8}+610311246440y^{2}wt^{9}-36685546478y^{2}t^{10}+941270482944z^{2}w^{10}-5201778941952z^{2}w^{9}t+12806421746688z^{2}w^{8}t^{2}-17996144451840z^{2}w^{7}t^{3}+15342136201728z^{2}w^{6}t^{4}-7764586159104z^{2}w^{5}t^{5}+1959215986368z^{2}w^{4}t^{6}+37421197560z^{2}w^{3}t^{7}-167614659612z^{2}w^{2}t^{8}+42158813256z^{2}wt^{9}-3571117746z^{2}t^{10}+10731143168zw^{11}-206484480000zw^{10}t+337639508992zw^{9}t^{2}+1078711663872zw^{8}t^{3}-4612880262912zw^{7}t^{4}+7500389382144zw^{6}t^{5}-6954373488384zw^{5}t^{6}+4041796605072zw^{4}t^{7}-1502641152804zw^{3}t^{8}+347949866840zw^{2}t^{9}-45796637985zwt^{10}+2620396177zt^{11}-930539339776w^{12}+5622808117248w^{11}t-15795398862848w^{10}t^{2}+26992435478016w^{9}t^{3}-30808182135552w^{8}t^{4}+24383801952000w^{7}t^{5}-13531078347264w^{6}t^{6}+5224355631312w^{5}t^{7}-1364861667072w^{4}t^{8}+227481210284w^{3}t^{9}-21413827968w^{2}t^{10}+834837304wt^{11}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 56.96.5.bp.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{14}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{7}+6X^{6}Y+13X^{5}Y^{2}+112X^{5}Z^{2}+4X^{4}Y^{3}+182X^{4}YZ^{2}+4X^{3}Y^{4}+140X^{3}Y^{2}Z^{2}+196X^{3}Z^{4}+84X^{2}Y^{3}Z^{2}+392X^{2}YZ^{4}+28XY^{4}Z^{2}+196XY^{2}Z^{4}+14Y^{5}Z^{2} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 28.96.2-28.c.1.8 | $28$ | $2$ | $2$ | $2$ | $0$ | $1^{3}$ |
| 56.24.0-56.bb.1.16 | $56$ | $8$ | $8$ | $0$ | $0$ | full Jacobian |
| 56.96.2-28.c.1.22 | $56$ | $2$ | $2$ | $2$ | $0$ | $1^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.384.9-56.y.1.12 | $56$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 56.384.9-56.y.2.12 | $56$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 56.384.9-56.y.3.16 | $56$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 56.384.9-56.y.4.16 | $56$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 56.384.9-56.ba.1.12 | $56$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 56.384.9-56.ba.2.12 | $56$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 56.384.9-56.ba.3.16 | $56$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 56.384.9-56.ba.4.16 | $56$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 56.384.11-56.bi.1.15 | $56$ | $2$ | $2$ | $11$ | $3$ | $1^{6}$ |
| 56.384.11-56.bs.1.11 | $56$ | $2$ | $2$ | $11$ | $2$ | $1^{6}$ |
| 56.384.11-56.dn.1.2 | $56$ | $2$ | $2$ | $11$ | $1$ | $1^{6}$ |
| 56.384.11-56.do.1.11 | $56$ | $2$ | $2$ | $11$ | $2$ | $1^{6}$ |
| 56.384.11-56.du.1.10 | $56$ | $2$ | $2$ | $11$ | $2$ | $1^{6}$ |
| 56.384.11-56.dx.1.21 | $56$ | $2$ | $2$ | $11$ | $3$ | $1^{6}$ |
| 56.384.11-56.ex.1.6 | $56$ | $2$ | $2$ | $11$ | $2$ | $1^{6}$ |
| 56.384.11-56.ey.1.13 | $56$ | $2$ | $2$ | $11$ | $1$ | $1^{6}$ |
| 56.384.11-56.fp.1.12 | $56$ | $2$ | $2$ | $11$ | $0$ | $2^{3}$ |
| 56.384.11-56.fp.2.12 | $56$ | $2$ | $2$ | $11$ | $0$ | $2^{3}$ |
| 56.384.11-56.fp.3.16 | $56$ | $2$ | $2$ | $11$ | $0$ | $2^{3}$ |
| 56.384.11-56.fp.4.16 | $56$ | $2$ | $2$ | $11$ | $0$ | $2^{3}$ |
| 56.384.11-56.fr.1.12 | $56$ | $2$ | $2$ | $11$ | $2$ | $2^{3}$ |
| 56.384.11-56.fr.2.12 | $56$ | $2$ | $2$ | $11$ | $2$ | $2^{3}$ |
| 56.384.11-56.fr.3.16 | $56$ | $2$ | $2$ | $11$ | $2$ | $2^{3}$ |
| 56.384.11-56.fr.4.16 | $56$ | $2$ | $2$ | $11$ | $2$ | $2^{3}$ |
| 56.576.13-56.cz.1.32 | $56$ | $3$ | $3$ | $13$ | $0$ | $2^{4}$ |
| 56.576.13-56.cz.2.32 | $56$ | $3$ | $3$ | $13$ | $0$ | $2^{4}$ |
| 56.576.13-56.dh.1.24 | $56$ | $3$ | $3$ | $13$ | $0$ | $1^{8}$ |
| 56.1344.41-56.np.1.24 | $56$ | $7$ | $7$ | $41$ | $11$ | $1^{22}\cdot2^{7}$ |
| 168.384.9-168.bzt.1.7 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bzt.2.7 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bzt.3.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bzt.4.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bzv.1.7 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bzv.2.7 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bzv.3.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bzv.4.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.11-168.ht.1.31 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.hv.1.30 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.if.1.31 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.ih.1.30 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.lm.1.31 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.lp.1.30 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.lz.1.31 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.ma.1.30 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.nx.1.15 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.nx.2.15 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.nx.3.13 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.nx.4.13 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.nz.1.15 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.nz.2.15 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.nz.3.13 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 168.384.11-168.nz.4.13 | $168$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.9-280.cy.1.16 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.384.9-280.cy.2.16 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.384.9-280.cy.3.32 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.384.9-280.cy.4.32 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.384.9-280.da.1.16 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.384.9-280.da.2.16 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.384.9-280.da.3.32 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.384.9-280.da.4.32 | $280$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.384.11-280.hv.1.4 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.hx.1.31 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.ih.1.5 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.ij.1.31 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.lm.1.8 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.lp.1.31 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.lz.1.13 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.ma.1.31 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.nx.1.16 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.nx.2.16 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.nx.3.32 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.nx.4.32 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.nz.1.16 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.nz.2.16 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.nz.3.32 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 280.384.11-280.nz.4.32 | $280$ | $2$ | $2$ | $11$ | $?$ | not computed |