Invariants
| Level: | $28$ | $\SL_2$-level: | $28$ | Newform level: | $196$ | ||
| Index: | $168$ | $\PSL_2$-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: | 28.168.9.1 |
Level structure
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 between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 7t$ |
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} $ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(4)$ | $4$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(4)$ | $4$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| 14.84.3.a.1 | $14$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 28.336.17.i.1 | $28$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 28.336.17.j.1 | $28$ | $2$ | $2$ | $17$ | $6$ | $1^{8}$ |
| 28.336.17.m.1 | $28$ | $2$ | $2$ | $17$ | $0$ | $1^{8}$ |
| 28.336.17.n.1 | $28$ | $2$ | $2$ | $17$ | $5$ | $1^{8}$ |
| 28.336.21.b.1 | $28$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 28.336.21.p.1 | $28$ | $2$ | $2$ | $21$ | $5$ | $1^{6}\cdot2^{3}$ |
| 28.336.21.s.1 | $28$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 28.336.21.t.1 | $28$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 28.336.21.w.1 | $28$ | $2$ | $2$ | $21$ | $5$ | $1^{10}\cdot2$ |
| 28.336.21.x.1 | $28$ | $2$ | $2$ | $21$ | $2$ | $1^{10}\cdot2$ |
| 28.336.21.ba.1 | $28$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 28.336.21.bb.1 | $28$ | $2$ | $2$ | $21$ | $3$ | $1^{10}\cdot2$ |
| 28.504.25.g.1 | $28$ | $3$ | $3$ | $25$ | $1$ | $1^{10}\cdot2^{3}$ |
| 56.336.17.be.1 | $56$ | $2$ | $2$ | $17$ | $6$ | $1^{8}$ |
| 56.336.17.bh.1 | $56$ | $2$ | $2$ | $17$ | $0$ | $1^{8}$ |
| 56.336.17.bq.1 | $56$ | $2$ | $2$ | $17$ | $5$ | $1^{8}$ |
| 56.336.17.bt.1 | $56$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 56.336.21.j.1 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 56.336.21.bu.1 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{6}\cdot2^{3}$ |
| 56.336.21.cc.1 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.336.21.cf.1 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.336.21.ci.1 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 56.336.21.cj.1 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 56.336.21.ck.1 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.336.21.cl.1 | $56$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 56.336.21.cm.1 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{10}\cdot2$ |
| 56.336.21.cn.1 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.336.21.co.1 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.336.21.cp.1 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{10}\cdot2$ |
| 56.336.21.cq.1 | $56$ | $2$ | $2$ | $21$ | $2$ | $1^{10}\cdot2$ |
| 56.336.21.cr.1 | $56$ | $2$ | $2$ | $21$ | $7$ | $1^{10}\cdot2$ |
| 56.336.21.cs.1 | $56$ | $2$ | $2$ | $21$ | $3$ | $1^{10}\cdot2$ |
| 56.336.21.ct.1 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{10}\cdot2$ |
| 56.336.21.cu.1 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 56.336.21.cv.1 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.336.21.cw.1 | $56$ | $2$ | $2$ | $21$ | $5$ | $1^{6}\cdot2^{3}$ |
| 56.336.21.cx.1 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{6}\cdot2^{3}$ |
| 56.336.21.de.1 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.336.21.dh.1 | $56$ | $2$ | $2$ | $21$ | $7$ | $1^{10}\cdot2$ |
| 56.336.21.dq.1 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{10}\cdot2$ |
| 56.336.21.dt.1 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{10}\cdot2$ |
| 84.336.17.i.1 | $84$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 84.336.17.j.1 | $84$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 84.336.17.m.1 | $84$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 84.336.17.n.1 | $84$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 84.336.21.s.1 | $84$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 84.336.21.t.1 | $84$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 84.336.21.w.1 | $84$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 84.336.21.x.1 | $84$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 84.336.21.ba.1 | $84$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 84.336.21.bb.1 | $84$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 84.336.21.be.1 | $84$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 84.336.21.bf.1 | $84$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 140.336.17.i.1 | $140$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 140.336.17.j.1 | $140$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 140.336.17.m.1 | $140$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 140.336.17.n.1 | $140$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 140.336.21.s.1 | $140$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 140.336.21.t.1 | $140$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 140.336.21.w.1 | $140$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 140.336.21.x.1 | $140$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 140.336.21.ba.1 | $140$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 140.336.21.bb.1 | $140$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 140.336.21.be.1 | $140$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 140.336.21.bf.1 | $140$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.17.be.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.336.17.bh.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.336.17.bq.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.336.17.bt.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.336.21.cc.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.cf.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.co.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.cr.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.cu.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.cv.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.cw.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.cx.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.cy.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.cz.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.da.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.db.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.dc.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.dd.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.de.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.df.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.dg.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.dh.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.di.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.dj.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.dq.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.dt.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.ec.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 168.336.21.ef.1 | $168$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.17.be.1 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.336.17.bh.1 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.336.17.bq.1 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.336.17.bt.1 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.336.21.cc.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.cf.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.co.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.cr.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.cu.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.cv.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.cw.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.cx.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.cy.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.cz.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.da.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.db.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.dc.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.dd.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.de.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.df.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.dg.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.dh.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.di.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.dj.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.dq.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.dt.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.ec.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 280.336.21.ef.1 | $280$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 308.336.17.i.1 | $308$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 308.336.17.j.1 | $308$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 308.336.17.m.1 | $308$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 308.336.17.n.1 | $308$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 308.336.21.s.1 | $308$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 308.336.21.t.1 | $308$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 308.336.21.w.1 | $308$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 308.336.21.x.1 | $308$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 308.336.21.ba.1 | $308$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 308.336.21.bb.1 | $308$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 308.336.21.be.1 | $308$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 308.336.21.bf.1 | $308$ | $2$ | $2$ | $21$ | $?$ | not computed |