Invariants
| Level: | $60$ | $\SL_2$-level: | $30$ | Newform level: | $900$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $5 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $6^{2}\cdot30^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ 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: | 30C5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.144.5.1258 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}4&55\\11&16\end{bmatrix}$, $\begin{bmatrix}16&35\\23&8\end{bmatrix}$, $\begin{bmatrix}16&45\\57&58\end{bmatrix}$, $\begin{bmatrix}56&35\\19&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.72.5.ba.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $15360$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{10}\cdot5^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 45.2.b.a, 900.2.a.b, 900.2.a.g$^{2}$ |
Models
Embedded model Embedded model in $\mathbb{P}^{6}$
| $ 0 $ | $=$ | $ - t v^{2} + u^{2} v $ |
| $=$ | $ - t u v + u^{3}$ | |
| $=$ | $ - t^{2} v + t u^{2}$ | |
| $=$ | $ - w t v + w u^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{7} + 5 x^{6} y + 5 x^{5} y^{2} - 189 x z^{6} - 675 y z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{6} + 1\right) y $ | $=$ | $ 156x^{12} - 743x^{6} + 911 $ |
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.
| Embedded model |
|---|
| $(0:0:0:0:0:0:1)$, $(0:0:0:1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{5^2}\cdot\frac{84348z^{2}w^{5}+168615zw^{6}+16830zw^{4}tv+57804zw^{2}t^{2}v^{2}-328503zw^{2}uv^{3}-93225ztuv^{4}+754975zv^{6}-135w^{7}-168912w^{5}tv+240195w^{3}t^{2}v^{2}-1210200w^{3}uv^{3}-335500wtuv^{4}+2696250wv^{6}}{v^{6}(7z+25w)}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 30.72.5.ba.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{7}+5X^{6}Y+5X^{5}Y^{2}-189XZ^{6}-675YZ^{6} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 30.72.5.ba.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 67z^{6}+135z^{5}w-13t^{6}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -z$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.48.1-30.i.2.4 | $60$ | $3$ | $3$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 60.72.2-15.a.2.8 | $60$ | $2$ | $2$ | $2$ | $0$ | $1^{3}$ |
| 60.72.2-15.a.2.15 | $60$ | $2$ | $2$ | $2$ | $0$ | $1^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.288.9-30.bd.1.3 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 60.288.9-30.bj.2.3 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 60.288.9-30.bp.1.3 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 60.288.9-30.bu.1.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 60.288.9-60.gn.1.1 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 60.288.9-60.hv.1.4 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 60.288.9-60.jc.1.3 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 60.288.9-60.kh.1.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 60.432.13-30.y.1.3 | $60$ | $3$ | $3$ | $13$ | $2$ | $1^{4}\cdot2^{2}$ |
| 60.576.21-60.hm.2.1 | $60$ | $4$ | $4$ | $21$ | $5$ | $1^{8}\cdot2^{4}$ |
| 60.720.25-30.eu.1.5 | $60$ | $5$ | $5$ | $25$ | $4$ | $1^{8}\cdot2^{6}$ |
| 120.288.9-120.rzr.1.5 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.saf.1.9 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.sjf.1.9 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.sjt.1.3 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.tdu.1.11 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.tei.1.5 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.tmw.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.tnk.1.9 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.432.17-90.bf.1.3 | $180$ | $3$ | $3$ | $17$ | $?$ | not computed |