Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $3600$ | ||
| Index: | $60$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $4 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $10^{2}\cdot20^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20A4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.60.4.41 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}13&54\\53&47\end{bmatrix}$, $\begin{bmatrix}21&2\\55&43\end{bmatrix}$, $\begin{bmatrix}29&52\\46&31\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $768$ |
| Full 60-torsion field degree: | $36864$ |
Jacobian
| Conductor: | $2^{12}\cdot3^{4}\cdot5^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 50.2.a.b, 200.2.a.e, 3600.2.a.l$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 105 x^{2} - 15 y^{2} + z^{2} + w^{2} $ |
| $=$ | $15 x^{3} - x z^{2} - x w^{2} - y z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{6} - 4 x^{4} y^{2} + x^{2} y^{4} + 105 x^{2} y^{2} z^{2} - 60 y^{4} z^{2} + 900 y^{2} z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{30}z$ |
Maps to other modular curves
$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{50856225xyz^{7}w+139227900xyz^{5}w^{3}+62588400xyz^{3}w^{5}+4905600xyzw^{7}+4251000y^{2}z^{8}+42634200y^{2}z^{6}w^{2}+44412000y^{2}z^{4}w^{4}+8846400y^{2}z^{2}w^{6}+206400y^{2}w^{8}-254728z^{10}-1432345z^{8}w^{2}-2440420z^{6}w^{4}-1618000z^{4}w^{6}-368320z^{2}w^{8}-13312w^{10}}{35280xyz^{7}w+664545xyz^{5}w^{3}+1179360xyz^{3}w^{5}+181440xyzw^{7}+960y^{2}z^{8}+83640y^{2}z^{6}w^{2}+454200y^{2}z^{4}w^{4}+256320y^{2}z^{2}w^{6}+7680y^{2}w^{8}-64z^{10}-3736z^{8}w^{2}-15353z^{6}w^{4}-20264z^{4}w^{6}-9088z^{2}w^{8}-512w^{10}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.30.2.k.1 | $20$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 60.12.0.bi.1 | $60$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 60.30.2.e.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 60.30.2.f.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.180.10.ft.1 | $60$ | $3$ | $3$ | $10$ | $2$ | $1^{6}$ |
| 60.180.14.hl.1 | $60$ | $3$ | $3$ | $14$ | $7$ | $1^{10}$ |
| 60.240.13.ru.1 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{9}$ |
| 60.240.17.mp.1 | $60$ | $4$ | $4$ | $17$ | $1$ | $1^{13}$ |
| 120.120.9.ca.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.120.9.cc.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.120.9.wf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.120.9.wg.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.120.9.bir.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.120.9.bis.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.120.9.bjg.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.120.9.bji.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |