Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $3600$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot4\cdot5^{2}\cdot20$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.36.1.43 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&5\\36&17\end{bmatrix}$, $\begin{bmatrix}1&20\\40&19\end{bmatrix}$, $\begin{bmatrix}3&35\\50&51\end{bmatrix}$, $\begin{bmatrix}39&5\\22&59\end{bmatrix}$, $\begin{bmatrix}59&50\\48&53\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $8$ |
| Cyclic 60-torsion field degree: | $128$ |
| Full 60-torsion field degree: | $61440$ |
Jacobian
| Conductor: | $2^{4}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3600.2.a.be |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 825x + 9250 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^2\cdot5^2}\cdot\frac{270x^{2}y^{10}-11738418750x^{2}y^{8}z^{2}+27407818884375000x^{2}y^{6}z^{4}-1635399787541660156250x^{2}y^{4}z^{6}+10433032598370563964843750x^{2}y^{2}z^{8}-34234038063627935943603515625x^{2}z^{10}-186975xy^{10}z+1732018950000xy^{8}z^{3}-1588707719455078125xy^{6}z^{5}+15291188325771679687500xy^{4}z^{7}-307541726023668804931640625xy^{2}z^{9}-402752757253661265563964843750xz^{11}-y^{12}+37995750y^{10}z^{2}-279136917281250y^{8}z^{4}+73914104310996093750y^{6}z^{6}-1389260964383215576171875y^{4}z^{8}+11666830021876909973144531250y^{2}z^{10}-45705804675457278247833251953125z^{12}}{z(11250x^{2}y^{8}z-2215856250x^{2}y^{6}z^{3}-267176221875000x^{2}y^{4}z^{5}+8287434885937500000x^{2}y^{2}z^{7}-15062615655468750000000x^{2}z^{9}+xy^{10}+967500xy^{8}z^{2}+202802484375xy^{6}z^{4}-10153711335937500xy^{4}z^{6}-77732241996093750000xy^{2}z^{8}+2067017972115234375000000xz^{10}-230y^{10}z-66121875y^{8}z^{3}+4784480156250y^{6}z^{5}+103609979296875000y^{4}z^{7}-3711195050976562500000y^{2}z^{9}+22176441286699218750000000z^{11})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_0(10)$ | $10$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.6.0.b.1 | $60$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
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.ch.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.ch.2 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.ci.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.ci.2 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.co.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.co.2 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.cp.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.cp.2 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.3.c.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.cn.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.oo.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.72.3.op.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.qz.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.qz.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.ra.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.ra.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.rh.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.rh.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.ri.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.ri.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.rn.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.72.3.ro.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.rv.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.72.3.rw.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.108.7.k.1 | $60$ | $3$ | $3$ | $7$ | $1$ | $1^{6}$ |
| 60.144.7.lz.1 | $60$ | $4$ | $4$ | $7$ | $4$ | $1^{6}$ |
| 60.180.7.ca.1 | $60$ | $5$ | $5$ | $7$ | $3$ | $1^{6}$ |
| 120.72.1.hw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.hw.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.hz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.hz.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.iu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.iu.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.ix.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.ix.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.3.he.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.qe.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dqs.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dqv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eec.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eec.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eef.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eef.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.efa.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.efa.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.efd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.efd.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.efs.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.efv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.egq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.egt.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 300.180.7.e.1 | $300$ | $5$ | $5$ | $7$ | $?$ | not computed |