Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $1800$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12L1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.36.1.9 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}11&42\\0&23\end{bmatrix}$, $\begin{bmatrix}27&8\\47&51\end{bmatrix}$, $\begin{bmatrix}27&52\\28&39\end{bmatrix}$, $\begin{bmatrix}35&18\\24&53\end{bmatrix}$, $\begin{bmatrix}39&52\\8&33\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $61440$ |
Jacobian
| Conductor: | $2^{3}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1800.2.a.m |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 975x - 8750 $ |
Rational points
This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^3\cdot5^3}\cdot\frac{90x^{2}y^{10}-74013750x^{2}y^{8}z^{2}-66065169375000x^{2}y^{6}z^{4}+58045567195371093750x^{2}y^{4}z^{6}+8195605357906867675781250x^{2}y^{2}z^{8}+180727310351072778167724609375x^{2}z^{10}-1575xy^{10}z-918337500xy^{8}z^{3}-2864811955078125xy^{6}z^{5}+4208440244558203125000xy^{4}z^{7}+352965120998660540771484375xy^{2}z^{9}+6327110795663672218322753906250xz^{11}-y^{12}+2432250y^{10}z^{2}-2142786656250y^{8}z^{4}+622154539300781250y^{6}z^{6}+184303030860731689453125y^{4}z^{8}+6728687641460906433105468750y^{2}z^{10}+45198406010875319103240966796875z^{12}}{z^{2}y^{6}(30x^{2}y^{2}-455625x^{2}z^{2}+375xy^{2}z+4556250xz^{3}-y^{4}-39750y^{2}z^{2}+398671875z^{4})}$ |
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 |
|---|---|---|---|---|---|---|
| 6.18.0.b.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.18.0.i.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.18.1.h.1 | $60$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 60.72.1.ek.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.el.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.eo.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.ep.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.3.bh.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.cv.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.gg.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.gi.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.ld.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.72.3.le.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.72.3.lh.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.li.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.lv.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.lw.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.md.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.me.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.180.13.lo.1 | $60$ | $5$ | $5$ | $13$ | $5$ | $1^{12}$ |
| 60.216.13.ob.1 | $60$ | $6$ | $6$ | $13$ | $4$ | $1^{12}$ |
| 60.360.25.cam.1 | $60$ | $10$ | $10$ | $25$ | $8$ | $1^{24}$ |
| 120.72.1.on.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.oq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.oz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.pc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.3.ks.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.se.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bod.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bor.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.daj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dam.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dav.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.day.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.deg.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.den.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dgk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dgr.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.108.5.a.1 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 180.108.5.bh.1 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |