Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $600$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12L1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.36.1.51 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}19&52\\34&35\end{bmatrix}$, $\begin{bmatrix}47&30\\57&43\end{bmatrix}$, $\begin{bmatrix}55&42\\3&7\end{bmatrix}$, $\begin{bmatrix}59&32\\32&13\end{bmatrix}$ |
| 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: | $61440$ |
Jacobian
| Conductor: | $2^{3}\cdot3\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 600.2.a.h |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + y^{2} - y z - z^{2} - w^{2} $ |
| $=$ | $2 x^{2} - y w - 2 z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 5 x^{2} y z + x^{2} z^{2} + 5 y^{2} z^{2} - 4 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 embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{12160000yz^{8}-18400000yz^{7}w+19192000yz^{6}w^{2}-9664000yz^{5}w^{3}+3476800yz^{4}w^{4}-377200yz^{3}w^{5}+45460yz^{2}w^{6}+8960yzw^{7}+1885yw^{8}+7520000z^{9}-15120000z^{8}w+21624000z^{7}w^{2}-16044000z^{6}w^{3}+8895600z^{5}w^{4}-2351400z^{4}w^{5}+478620z^{3}w^{6}+30690z^{2}w^{7}+9510zw^{8}+2414w^{9}}{w^{6}(80yz^{2}-20yzw+5yw^{2}+60z^{3}-30z^{2}w+30zw^{2}-2w^{3})}$ |
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 |
|---|---|---|---|---|---|---|
| 4.6.0.d.1 | $4$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 15.6.0.b.1 | $15$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.18.0.k.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 30.18.0.b.1 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.18.1.j.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.72.3.w.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.cf.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.qg.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.qh.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.wp.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.wq.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.xn.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.xo.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.180.13.ng.1 | $60$ | $5$ | $5$ | $13$ | $4$ | $1^{12}$ |
| 60.216.13.pt.1 | $60$ | $6$ | $6$ | $13$ | $1$ | $1^{12}$ |
| 60.360.25.cce.1 | $60$ | $10$ | $10$ | $25$ | $6$ | $1^{24}$ |
| 120.72.3.lw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.oi.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eba.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ebh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.fka.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.fkh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.fqm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.fqt.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.grp.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.grq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gvl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gvm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.hll.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.hlm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.hmq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.hms.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.5.ei.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ek.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.wn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.wo.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.byr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bys.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cef.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ceg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.108.5.cf.1 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 180.324.21.bw.1 | $180$ | $9$ | $9$ | $21$ | $?$ | not computed |