Invariants
| Level: | $60$ | $\SL_2$-level: | $30$ | Newform level: | $720$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $6^{2}\cdot30^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $16$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.263 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}7&25\\53&22\end{bmatrix}$, $\begin{bmatrix}17&40\\23&23\end{bmatrix}$, $\begin{bmatrix}39&50\\47&9\end{bmatrix}$, $\begin{bmatrix}59&10\\44&23\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $192$ |
| Full 60-torsion field degree: | $30720$ |
Jacobian
| Conductor: | $2^{4}\cdot3^{2}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 720.2.a.h |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 12x + 11 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(1:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^{15}}\cdot\frac{18x^{2}y^{22}+5751x^{2}y^{20}z^{2}+656100x^{2}y^{18}z^{4}-3720087x^{2}y^{16}z^{6}-5548244040x^{2}y^{14}z^{8}+197713589553x^{2}y^{12}z^{10}+43257047278806x^{2}y^{10}z^{12}-2321727695171865x^{2}y^{8}z^{14}-83849375367698166x^{2}y^{6}z^{16}+10069448966975633760x^{2}y^{4}z^{18}-306530305680249241902x^{2}y^{2}z^{20}+3329165719614239170056x^{2}z^{22}+99xy^{22}z+16929xy^{20}z^{3}-65610xy^{18}z^{5}+55269864xy^{16}z^{7}+33050315790xy^{14}z^{9}-42874534116xy^{12}z^{11}-234653616618498xy^{10}z^{13}+8272674934124580xy^{8}z^{15}+550541872469168667xy^{6}z^{17}-47115783516652652745xy^{4}z^{19}+1283594496898425668064xy^{2}z^{21}-12831023347362666190944xz^{23}+y^{24}+504y^{22}z^{2}+69903y^{20}z^{4}+11022480y^{18}z^{6}+920987253y^{16}z^{8}-79760791044y^{14}z^{10}-6348142992591y^{12}z^{12}+666924225948072y^{10}z^{14}+3462429211959015y^{8}z^{16}-2214670080597615576y^{6}z^{18}+95875401831948983106y^{4}z^{20}-1662818784587332632396y^{2}z^{22}+9502007722383724020009z^{24}}{z^{10}y^{6}(12x^{2}y^{6}-4131x^{2}y^{4}z^{2}+284310x^{2}y^{2}z^{4}-5255361x^{2}z^{6}-114xy^{6}z+22842xy^{4}z^{3}-1257525xy^{2}z^{5}+20253807xz^{7}-y^{8}+696y^{6}z^{2}-70470y^{4}z^{4}+2055780y^{2}z^{6}-14998446z^{8})}$ |
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 |
|---|---|---|---|---|---|---|
| 15.36.0.a.1 | $15$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.36.0.cg.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.36.1.gb.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 |
|---|---|---|---|---|---|---|
| 60.144.9.p.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.q.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.ed.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.ef.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.kb.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.kd.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.ki.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.kk.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.216.9.bv.2 | $60$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.288.13.ss.2 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 60.360.21.dg.1 | $60$ | $5$ | $5$ | $21$ | $5$ | $1^{8}\cdot2^{6}$ |
| 120.144.9.itf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.itm.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.nrl.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.nsg.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tlf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tma.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tnj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.toe.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.216.13.hv.1 | $180$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 300.360.21.y.1 | $300$ | $5$ | $5$ | $21$ | $?$ | not computed |