Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $3600$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12G3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.144.3.523 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}7&34\\20&43\end{bmatrix}$, $\begin{bmatrix}17&52\\46&19\end{bmatrix}$, $\begin{bmatrix}23&18\\18&7\end{bmatrix}$, $\begin{bmatrix}49&4\\28&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.72.3.t.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $15360$ |
Jacobian
| Conductor: | $2^{8}\cdot3^{5}\cdot5^{2}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}$ |
| Newforms: | 36.2.a.a$^{2}$, 1200.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ 2 x w + 2 x t + 4 x u + z t $ |
| $=$ | $6 x t + y w + y t + 2 y u$ | |
| $=$ | $5 y^{2} - 5 y z - 3 t^{2}$ | |
| $=$ | $6 x^{2} + 2 y z + 5 z^{2} - w^{2} + w t + 2 w u + 2 t^{2} + 2 t u + 2 u^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 625 x^{8} - 250 x^{6} y^{2} - 500 x^{6} y z - 1750 x^{6} z^{2} + 25 x^{4} y^{4} + 100 x^{4} y^{3} z + \cdots + 144 z^{8} $ |
Geometric Weierstrass model Geometric Weierstrass model
| $ 729 w^{2} $ | $=$ | $ -3625 x^{4} + 1000 x^{3} y + 725 x^{2} z^{2} + 5100 x y z^{2} + 1995 z^{4} $ |
| $0$ | $=$ | $2 x^{2} - 2 x y + 3 y^{2} - z^{2}$ |
Rational points
This modular curve has no $\Q_p$ points for $p=19$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{3745812xt^{8}+20540184xt^{7}u+46379664xt^{6}u^{2}+55821792xt^{5}u^{3}+40113600xt^{4}u^{4}+21566592xt^{3}u^{5}+5521152xt^{2}u^{6}+1577472xtu^{7}+49152yt^{8}+1248604yt^{7}u+4202092yt^{6}u^{2}+7038816yt^{5}u^{3}+6415312yt^{4}u^{4}+3660736yt^{3}u^{5}+1529664yt^{2}u^{6}+265216ytu^{7}+66304yu^{8}-536178zwt^{7}-1661532zwt^{6}u-1864392zwt^{5}u^{2}-1565424zwt^{4}u^{3}-482400zwt^{3}u^{4}-190272zwt^{2}u^{5}+2688zwtu^{6}+768zwu^{7}+1019605zt^{8}+3875544zt^{7}u+6197208zt^{6}u^{2}+4909696zt^{5}u^{3}+3121536zt^{4}u^{4}+802176zt^{3}u^{5}+270976zt^{2}u^{6}+3072ztu^{7}+768zu^{8}}{1740xt^{8}+20760xt^{7}u+102528xt^{6}u^{2}+249600xt^{5}u^{3}+314880xt^{4}u^{4}+211968xt^{3}u^{5}+86016xt^{2}u^{6}+24576xtu^{7}+580yt^{7}u+5380yt^{6}u^{2}+20352yt^{5}u^{3}+38080yt^{4}u^{4}+36352yt^{3}u^{5}+16896yt^{2}u^{6}+4096ytu^{7}+1024yu^{8}-414zwt^{7}-3132zwt^{6}u-8448zwt^{5}u^{2}-10752zwt^{4}u^{3}-7680zwt^{3}u^{4}-3072zwt^{2}u^{5}+451zt^{8}+5316zt^{7}u+20676zt^{6}u^{2}+34816zt^{5}u^{3}+27648zt^{4}u^{4}+12288zt^{3}u^{5}+4096zt^{2}u^{6}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.72.3.t.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 625X^{8}-250X^{6}Y^{2}+25X^{4}Y^{4}-500X^{6}YZ+100X^{4}Y^{3}Z-1750X^{6}Z^{2}+150X^{4}Y^{2}Z^{2}+100X^{4}YZ^{3}+1525X^{4}Z^{4}+60X^{2}Y^{2}Z^{4}+120X^{2}YZ^{5}-660X^{2}Z^{6}+144Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.72.2-12.d.1.3 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.1-60.a.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 60.72.1-60.a.1.4 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 60.72.2-12.d.1.5 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.72.2-60.f.1.6 | $60$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 60.72.2-60.f.1.15 | $60$ | $2$ | $2$ | $2$ | $1$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.288.7-60.m.1.1 | $60$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
| 60.288.7-60.m.1.4 | $60$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
| 60.288.7-60.n.1.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 60.288.7-60.n.1.6 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 60.288.7-60.dr.1.3 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 60.288.7-60.dr.1.5 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 60.288.7-60.ds.1.4 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 60.288.7-60.ds.1.7 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 60.720.27-60.bf.1.3 | $60$ | $5$ | $5$ | $27$ | $8$ | $1^{24}$ |
| 60.864.29-60.dr.1.6 | $60$ | $6$ | $6$ | $29$ | $11$ | $1^{26}$ |
| 60.1440.53-60.hr.1.3 | $60$ | $10$ | $10$ | $53$ | $17$ | $1^{50}$ |
| 120.288.7-120.ce.1.4 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ce.1.14 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ck.1.3 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ck.1.13 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ui.1.7 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.ui.1.9 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.up.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.up.1.10 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 180.432.15-180.g.1.5 | $180$ | $3$ | $3$ | $15$ | $?$ | not computed |