Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $3600$ | ||
| 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 | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20C7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.240.7.3 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}13&6\\34&7\end{bmatrix}$, $\begin{bmatrix}35&18\\42&23\end{bmatrix}$, $\begin{bmatrix}35&46\\36&25\end{bmatrix}$, $\begin{bmatrix}41&10\\56&29\end{bmatrix}$, $\begin{bmatrix}55&28\\8&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.120.7.a.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $9216$ |
Jacobian
| Conductor: | $2^{20}\cdot3^{8}\cdot5^{14}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 50.2.a.b$^{2}$, 100.2.a.a, 3600.2.a.bc, 3600.2.a.be, 3600.2.a.bf, 3600.2.a.h |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x t - 2 x v + y w - y t + y u + 2 y v + z u + 2 z v $ |
| $=$ | $x w + x t - x u - x v - y w + y t + z w - z t + z u + 2 z v$ | |
| $=$ | $3 x^{2} + 3 y z + 3 z^{2} - 2 w^{2} + w t - t^{2} - t v + u^{2} + u v + v^{2}$ | |
| $=$ | $3 x w - x t + 2 x u + x v - 2 y w - 2 y t - 3 z u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{8} y^{2} - 22 x^{7} y^{3} + 18 x^{7} y z^{2} + 191 x^{6} y^{4} - 249 x^{6} y^{2} z^{2} + \cdots + 1296 y^{2} z^{8} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 :
| $\displaystyle X$ | $=$ | $\displaystyle -x+3y-z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -3x-y+2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -2x+y-2z$ |
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 60.120.7.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{8}Y^{2}-22X^{7}Y^{3}+18X^{7}YZ^{2}+191X^{6}Y^{4}-249X^{6}Y^{2}Z^{2}+36X^{6}Z^{4}-816X^{5}Y^{5}+1182X^{5}Y^{3}Z^{2}-612X^{5}YZ^{4}+1719X^{4}Y^{6}-2160X^{4}Y^{4}Z^{2}+2844X^{4}Y^{2}Z^{4}-432X^{4}Z^{6}-1478X^{3}Y^{7}+720X^{3}Y^{5}Z^{2}-1476X^{3}Y^{3}Z^{4}+3456X^{3}YZ^{6}+109X^{2}Y^{8}+2253X^{2}Y^{6}Z^{2}-4572X^{2}Y^{4}Z^{4}-1728X^{2}Y^{2}Z^{6}+1296X^{2}Z^{8}+276XY^{9}-1536XY^{7}Z^{2}+3024XY^{5}Z^{4}+864XY^{3}Z^{6}-2592XYZ^{8}+36Y^{10}-420Y^{8}Z^{2}+1332Y^{6}Z^{4}-2160Y^{4}Z^{6}+1296Y^{2}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 |
| 12.24.0-12.a.1.2 | $12$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.24.0-12.a.1.2 | $12$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 20.120.3-10.a.1.4 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 60.120.3-10.a.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 |
|---|---|---|---|---|---|---|
| 60.480.13-60.a.1.2 | $60$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
| 60.480.13-60.c.1.3 | $60$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
| 60.480.13-60.i.1.1 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 60.480.13-60.k.1.2 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 60.480.13-60.y.1.1 | $60$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
| 60.480.13-60.ba.1.2 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 60.480.13-60.bg.1.4 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 60.480.13-60.bi.1.1 | $60$ | $2$ | $2$ | $13$ | $5$ | $1^{6}$ |
| 60.480.15-60.a.1.5 | $60$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 60.480.15-60.a.1.7 | $60$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 60.480.15-60.c.1.3 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.c.1.9 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.g.1.1 | $60$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 60.480.15-60.g.1.11 | $60$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 60.480.15-60.h.1.5 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.h.1.7 | $60$ | $2$ | $2$ | $15$ | $3$ | $1^{8}$ |
| 60.480.15-60.n.1.4 | $60$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 60.480.15-60.n.1.10 | $60$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 60.480.15-60.o.1.4 | $60$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 60.480.15-60.o.1.10 | $60$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 60.480.15-60.u.1.2 | $60$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 60.480.15-60.u.1.12 | $60$ | $2$ | $2$ | $15$ | $4$ | $1^{8}$ |
| 60.480.15-60.v.1.2 | $60$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 60.480.15-60.v.1.12 | $60$ | $2$ | $2$ | $15$ | $6$ | $1^{8}$ |
| 60.720.19-60.bc.1.1 | $60$ | $3$ | $3$ | $19$ | $6$ | $1^{12}$ |
| 60.720.27-60.cs.1.3 | $60$ | $3$ | $3$ | $27$ | $11$ | $1^{20}$ |
| 60.960.33-60.o.1.4 | $60$ | $4$ | $4$ | $33$ | $8$ | $1^{26}$ |
| 120.480.13-120.b.1.15 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.h.1.15 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.z.1.15 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bf.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.cv.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.db.1.15 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.dt.1.15 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.dz.1.15 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.15-120.b.1.14 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.b.1.32 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.f.1.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.f.1.28 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.o.1.14 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.o.1.32 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.r.1.16 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.r.1.28 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bx.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.bx.1.21 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ca.1.7 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ca.1.21 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cq.1.5 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.cq.1.23 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ct.1.5 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.480.15-120.ct.1.23 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |