Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $196$ | ||
| Index: | $336$ | $\PSL_2$-index: | $168$ | ||||
| Genus: | $9 = 1 + \frac{ 168 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $3$ are rational) | Cusp widths | $7^{8}\cdot28^{4}$ | Cusp orbits | $1^{3}\cdot3^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28A9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.336.9.7 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}17&7\\8&11\end{bmatrix}$, $\begin{bmatrix}29&41\\48&17\end{bmatrix}$, $\begin{bmatrix}31&45\\24&39\end{bmatrix}$, $\begin{bmatrix}37&10\\52&31\end{bmatrix}$, $\begin{bmatrix}37&22\\44&47\end{bmatrix}$, $\begin{bmatrix}49&16\\40&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.168.9.c.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $4$ |
| Cyclic 56-torsion field degree: | $96$ |
| Full 56-torsion field degree: | $9216$ |
Jacobian
| Conductor: | $2^{12}\cdot7^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2^{3}$ |
| Newforms: | 14.2.a.a$^{2}$, 98.2.a.b$^{2}$, 196.2.a.b, 196.2.a.c |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x u - y t - y u + z u + u r $ |
| $=$ | $x u - y t - y r - y s$ | |
| $=$ | $x u - x r - x s - z t - z r - z s - t r - r^{2} - r s$ | |
| $=$ | $2 x u + x r + x s + y t + z u - w t - w u - w r - w s - t v + t r - u v - v r - v s + r^{2} + r s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 14700 x^{16} - 380240 x^{15} y + 426888 x^{14} y^{2} + 3724 x^{14} z^{2} + 11458944 x^{13} y^{3} + \cdots + 400 y^{10} z^{6} $ |
Rational points
This modular curve has 3 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(2:1:0:1:0:0:1:-1:1)$, $(0:0:-1/5:0:-2/5:0:-1/5:-3/5:1)$, $(0:0:-1:0:2:0:-1:-3:1)$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 14.84.3.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle t+5u+r+s$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -4t-6u+3r+3s$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t-2u+r+s$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}Y^{2}+X^{3}Z+4XY^{2}Z+Y^{3}Z+2X^{2}Z^{2}-XYZ^{2}-2XZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 28.168.9.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 7t$ |
Equation of the image curve:
| $0$ | $=$ | $ -14700X^{16}-380240X^{15}Y+426888X^{14}Y^{2}+3724X^{14}Z^{2}+11458944X^{13}Y^{3}+28616X^{13}YZ^{2}-41070820X^{12}Y^{4}+110740X^{12}Y^{2}Z^{2}-196X^{12}Z^{4}+35158872X^{11}Y^{5}+1032332X^{11}Y^{3}Z^{2}-2352X^{11}YZ^{4}+33691861X^{10}Y^{6}-359660X^{10}Y^{4}Z^{2}-17640X^{10}Y^{2}Z^{4}+4X^{10}Z^{6}-48383776X^{9}Y^{7}-15772904X^{9}Y^{5}Z^{2}-100548X^{9}Y^{3}Z^{4}+88X^{9}YZ^{6}-16456356X^{8}Y^{8}+20848128X^{8}Y^{6}Z^{2}-336532X^{8}Y^{4}Z^{4}+636X^{8}Y^{2}Z^{6}+19492935X^{7}Y^{9}-1731464X^{7}Y^{7}Z^{2}-837508X^{7}Y^{5}Z^{4}+1144X^{7}Y^{3}Z^{6}+15354052X^{6}Y^{10}+2867872X^{6}Y^{8}Z^{2}-1032773X^{6}Y^{6}Z^{4}-4788X^{6}Y^{4}Z^{6}-4479237X^{5}Y^{11}+12649252X^{5}Y^{9}Z^{2}+2454900X^{5}Y^{7}Z^{4}-14616X^{5}Y^{5}Z^{6}-4066951X^{4}Y^{12}-8650264X^{4}Y^{10}Z^{2}-1922956X^{4}Y^{8}Z^{4}+10248X^{4}Y^{6}Z^{6}-916839X^{3}Y^{13}+1934912X^{3}Y^{11}Z^{2}+565558X^{3}Y^{9}Z^{4}+29504X^{3}Y^{7}Z^{6}+214914X^{2}Y^{14}-30772X^{2}Y^{12}Z^{2}-32144X^{2}Y^{10}Z^{4}+5956X^{2}Y^{8}Z^{6}-15876XY^{15}+7056XY^{13}Z^{2}-22540XY^{11}Z^{4}-4240XY^{9}Z^{6}+3528Y^{16}-3920Y^{14}Z^{2}-1225Y^{12}Z^{4}+400Y^{10}Z^{6} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.12.0-4.c.1.6 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $12$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.12.0-4.c.1.6 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.672.17-28.i.1.5 | $56$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 56.672.17-28.j.1.2 | $56$ | $2$ | $2$ | $17$ | $6$ | $1^{8}$ |
| 56.672.17-28.m.1.5 | $56$ | $2$ | $2$ | $17$ | $0$ | $1^{8}$ |
| 56.672.17-28.n.1.2 | $56$ | $2$ | $2$ | $17$ | $5$ | $1^{8}$ |
| 56.672.21-28.b.1.37 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-28.p.1.21 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-28.s.1.11 | $56$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-28.t.1.11 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-28.w.1.8 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{10}\cdot2$ |
| 56.672.21-28.x.1.8 | $56$ | $2$ | $2$ | $21$ | $2$ | $1^{10}\cdot2$ |
| 56.672.21-28.ba.1.7 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.672.21-28.bb.1.7 | $56$ | $2$ | $2$ | $21$ | $3$ | $1^{10}\cdot2$ |
| 56.1008.25-28.g.1.5 | $56$ | $3$ | $3$ | $25$ | $1$ | $1^{10}\cdot2^{3}$ |
| 56.672.17-56.be.1.16 | $56$ | $2$ | $2$ | $17$ | $6$ | $1^{8}$ |
| 56.672.17-56.bh.1.16 | $56$ | $2$ | $2$ | $17$ | $0$ | $1^{8}$ |
| 56.672.17-56.bq.1.16 | $56$ | $2$ | $2$ | $17$ | $5$ | $1^{8}$ |
| 56.672.17-56.bt.1.16 | $56$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 56.672.21-56.j.1.15 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.bu.1.15 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cc.1.15 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cf.1.15 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.ci.1.16 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.ci.1.32 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cj.1.32 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cj.1.48 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.ck.1.28 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.ck.1.32 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cl.1.24 | $56$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cl.1.32 | $56$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cm.1.16 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{10}\cdot2$ |
| 56.672.21-56.cm.1.32 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{10}\cdot2$ |
| 56.672.21-56.cn.1.16 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.672.21-56.cn.1.32 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.672.21-56.co.1.16 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.672.21-56.co.1.32 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.672.21-56.cp.1.16 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{10}\cdot2$ |
| 56.672.21-56.cp.1.32 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{10}\cdot2$ |
| 56.672.21-56.cq.1.31 | $56$ | $2$ | $2$ | $21$ | $2$ | $1^{10}\cdot2$ |
| 56.672.21-56.cq.1.32 | $56$ | $2$ | $2$ | $21$ | $2$ | $1^{10}\cdot2$ |
| 56.672.21-56.cr.1.31 | $56$ | $2$ | $2$ | $21$ | $7$ | $1^{10}\cdot2$ |
| 56.672.21-56.cr.1.32 | $56$ | $2$ | $2$ | $21$ | $7$ | $1^{10}\cdot2$ |
| 56.672.21-56.cs.1.24 | $56$ | $2$ | $2$ | $21$ | $3$ | $1^{10}\cdot2$ |
| 56.672.21-56.cs.1.32 | $56$ | $2$ | $2$ | $21$ | $3$ | $1^{10}\cdot2$ |
| 56.672.21-56.ct.1.28 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{10}\cdot2$ |
| 56.672.21-56.ct.1.32 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{10}\cdot2$ |
| 56.672.21-56.cu.1.16 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cu.1.32 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cv.1.16 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cv.1.32 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cw.1.16 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cw.1.32 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cx.1.16 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.cx.1.32 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.de.1.12 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.672.21-56.dh.1.12 | $56$ | $2$ | $2$ | $21$ | $7$ | $1^{10}\cdot2$ |
| 56.672.21-56.dq.1.12 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{10}\cdot2$ |
| 56.672.21-56.dt.1.12 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{10}\cdot2$ |