Invariants
| Level: | $60$ | $\SL_2$-level: | $10$ | 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 | $2^{3}\cdot10^{3}$ | 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: | 10G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.36.1.46 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&20\\14&57\end{bmatrix}$, $\begin{bmatrix}11&25\\28&33\end{bmatrix}$, $\begin{bmatrix}17&20\\52&57\end{bmatrix}$, $\begin{bmatrix}31&40\\42&13\end{bmatrix}$, $\begin{bmatrix}49&25\\6&17\end{bmatrix}$ |
| 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^5\cdot5^5}\cdot\frac{270x^{2}y^{10}+34931250x^{2}y^{8}z^{2}-1250690625000x^{2}y^{6}z^{4}+9829582558593750x^{2}y^{4}z^{6}-6888826019377441406250x^{2}y^{2}z^{8}-92981081570208892822265625x^{2}z^{10}-24975xy^{10}z-2454300000xy^{8}z^{3}-105091330078125xy^{6}z^{5}-1119866591601562500xy^{4}z^{7}-189392211770855712890625xy^{2}z^{9}+1627167695902088928222656250xz^{11}-y^{12}+735750y^{10}z^{2}+62792718750y^{8}z^{4}+4897555839843750y^{6}z^{6}+29523649440673828125y^{4}z^{8}-4305918540631530761718750y^{2}z^{10}+25569600379556774139404296875z^{12}}{z^{3}(6750x^{2}y^{6}z-265781250x^{2}y^{4}z^{3}-10495037109375x^{2}y^{2}z^{5}+70858800000000000x^{2}z^{7}+xy^{8}+270000xy^{6}z^{2}+19572890625xy^{4}z^{4}-288709628906250xy^{2}z^{6}-1240029000000000000xz^{8}-170y^{8}z-11390625y^{6}z^{3}+290650781250y^{4}z^{5}+527250673828125y^{2}z^{7}-19486170000000000000z^{9})}$ |
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.a.1 | $60$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 60.12.1.n.1 | $60$ | $3$ | $3$ | $1$ | $1$ | 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.cf.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.cf.2 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.cg.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.cg.2 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.cm.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.cm.2 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.cn.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.1.cn.2 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.72.3.fd.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.ff.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.or.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.ou.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.72.3.qx.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.qx.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.qy.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.qy.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.rf.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.rf.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.rg.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.rg.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 60.72.3.rq.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.72.3.rt.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.rv.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.72.3.rx.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.108.7.j.1 | $60$ | $3$ | $3$ | $7$ | $1$ | $1^{6}$ |
| 60.144.7.ly.1 | $60$ | $4$ | $4$ | $7$ | $4$ | $1^{6}$ |
| 60.180.7.by.1 | $60$ | $5$ | $5$ | $7$ | $3$ | $1^{6}$ |
| 120.72.1.hq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.hq.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.ht.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.ht.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.io.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.io.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.ir.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.ir.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.3.bgq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bgw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.drb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.drk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edw.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edz.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eeu.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eeu.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eex.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eex.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.egb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.egk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.egp.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.egv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 300.180.7.d.1 | $300$ | $5$ | $5$ | $7$ | $?$ | not computed |