Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $200$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $7 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $10^{4}\cdot20^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20C7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.240.7.13 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}17&44\\2&13\end{bmatrix}$, $\begin{bmatrix}41&52\\26&39\end{bmatrix}$, $\begin{bmatrix}47&0\\20&37\end{bmatrix}$, $\begin{bmatrix}49&20\\30&19\end{bmatrix}$, $\begin{bmatrix}55&56\\14&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.120.7.b.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $9216$ |
Jacobian
| Conductor: | $2^{13}\cdot5^{14}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 50.2.a.b$^{3}$, 100.2.a.a$^{2}$, 200.2.a.c, 200.2.a.e |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ 2 x^{2} - z^{2} + w v - t v - u v + v^{2} $ |
| $=$ | $x y + x z + y^{2} - 3 y z - t u - u v$ | |
| $=$ | $x^{2} - x y - x z + z^{2} + w^{2} - w t - w u + 2 w v + t u - t v - u^{2} + u v + v^{2}$ | |
| $=$ | $y^{2} + 2 y z - z^{2} - w v - t^{2} + 2 t u - 2 u v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 256 x^{12} - 1056 x^{10} y^{2} - 1184 x^{10} z^{2} + 321 x^{8} y^{4} + 4582 x^{8} y^{2} z^{2} + \cdots + y^{4} z^{8} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=3$, and therefore no rational points.
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 10.60.3.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x+2y-3z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x-4y+z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3x-y-z$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}-3X^{3}Y-5X^{2}Y^{2}-4XY^{3}-2Y^{4}+3X^{3}Z-18X^{2}YZ-17XY^{2}Z+4Y^{3}Z-5X^{2}Z^{2}+17XYZ^{2}-6Y^{2}Z^{2}+4XZ^{3}+4YZ^{3}-2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.120.7.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 256X^{12}-1056X^{10}Y^{2}-1184X^{10}Z^{2}+321X^{8}Y^{4}+4582X^{8}Y^{2}Z^{2}+1401X^{8}Z^{4}+1584X^{6}Y^{6}-1976X^{6}Y^{4}Z^{2}-4999X^{6}Y^{2}Z^{4}-74X^{6}Z^{6}+576X^{4}Y^{8}+256X^{4}Y^{6}Z^{2}+3950X^{4}Y^{4}Z^{4}+196X^{4}Y^{2}Z^{6}+X^{4}Z^{8}-1152X^{2}Y^{8}Z^{2}-1808X^{2}Y^{6}Z^{4}+8X^{2}Y^{4}Z^{6}-3X^{2}Y^{2}Z^{8}+576Y^{8}Z^{4}-32Y^{6}Z^{6}+Y^{4}Z^{8} $ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 12.24.0-4.b.1.1 | $12$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.24.0-4.b.1.1 | $12$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 60.120.3-10.a.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 60.120.3-20.c.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.480.13-20.b.1.2 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 60.480.13-20.d.1.6 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 60.480.13-60.f.1.1 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 60.480.13-60.h.1.6 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 60.480.13-20.j.1.5 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 60.480.13-20.l.1.1 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 60.480.13-60.bd.1.12 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 60.480.13-60.bf.1.2 | $60$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
| 60.480.15-20.b.1.3 | $60$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 60.480.15-60.c.1.15 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-20.e.1.3 | $60$ | $2$ | $2$ | $15$ | $1$ | $1^{8}$ |
| 60.480.15-60.f.1.7 | $60$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 60.480.15-20.h.1.4 | $60$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 60.480.15-20.k.1.8 | $60$ | $2$ | $2$ | $15$ | $2$ | $1^{8}$ |
| 60.480.15-60.m.1.2 | $60$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 60.480.15-60.t.1.4 | $60$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 60.720.19-20.k.1.8 | $60$ | $3$ | $3$ | $19$ | $1$ | $1^{12}$ |
| 60.720.27-60.cr.1.6 | $60$ | $3$ | $3$ | $27$ | $9$ | $1^{20}$ |
| 60.960.33-60.n.1.8 | $60$ | $4$ | $4$ | $33$ | $2$ | $1^{26}$ |
| 120.480.13-40.e.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-40.k.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.q.1.34 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.w.1.21 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-40.bc.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-40.bi.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.dk.1.34 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.dq.1.34 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.15-40.c.1.6 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.e.1.48 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.h.1.6 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.l.1.24 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.m.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.n.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.q.1.18 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.r.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.s.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.s.2.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.t.1.4 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.t.2.6 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.u.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.u.2.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.v.1.6 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.v.2.10 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.w.1.4 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.w.2.6 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.x.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.x.2.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.y.1.6 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.y.2.10 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.z.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.z.2.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bc.1.64 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bd.1.64 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.be.1.9 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bg.1.72 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bh.1.72 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bi.1.80 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bi.2.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bj.1.80 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bj.2.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bk.1.80 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bk.2.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.bl.1.5 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bl.1.80 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bl.2.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bm.1.80 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bm.2.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bn.1.80 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bn.2.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bo.1.80 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bo.2.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bp.1.80 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bp.2.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bu.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.cc.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.cd.1.18 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.cg.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-40.ch.1.3 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cn.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.eo.1.80 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ep.1.80 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.es.1.70 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.et.1.70 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.17-40.bu.1.14 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-40.bv.1.3 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-40.cg.1.3 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-40.ch.1.3 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-40.cw.1.3 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-40.cx.1.3 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-40.da.1.3 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-40.db.1.14 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.du.1.80 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.dv.1.80 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.go.1.40 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.gp.1.40 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.ic.1.40 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.id.1.40 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.ig.1.72 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.ih.1.72 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |