Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $3600$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.96.1.190 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}1&18\\36&19\end{bmatrix}$, $\begin{bmatrix}5&24\\57&55\end{bmatrix}$, $\begin{bmatrix}5&34\\18&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.48.1.q.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $192$ |
| Full 60-torsion field degree: | $23040$ |
Jacobian
| Conductor: | $2^{4}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3600.2.a.v |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - 4 x y + 3 x z + y^{2} - 3 y z $ |
| $=$ | $6 x^{2} + 26 x y + 13 x z + 6 y^{2} - 13 y z + 45 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} + 3 x^{2} y^{2} + 5 x^{2} z^{2} - 6 x y^{2} z + 3 y^{2} z^{2} + 5 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^4}{5^2}\cdot\frac{1249996267520xz^{11}-218752332800xz^{9}w^{2}-3519406080xz^{7}w^{4}+398391680xz^{5}w^{6}+12143200xz^{3}w^{8}+94380xzw^{10}-1249996267520yz^{11}+218752332800yz^{9}w^{2}+3519406080yz^{7}w^{4}-398391680yz^{5}w^{6}-12143200yz^{3}w^{8}-94380yzw^{10}+2249996267520z^{12}-456252519424z^{10}w^{2}+699215616z^{8}w^{4}+1089600128z^{6}w^{6}+23041040z^{4}w^{8}+44220z^{2}w^{10}-1331w^{12}}{w^{4}(2880xz^{7}+1720xz^{5}w^{2}+350xz^{3}w^{4}+30xzw^{6}-2880yz^{7}-1720yz^{5}w^{2}-350yz^{3}w^{4}-30yzw^{6}+2880z^{8}+1864z^{6}w^{2}+445z^{4}w^{4}+54z^{2}w^{6}+w^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.48.1.q.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ 5X^{4}+3X^{2}Y^{2}-6XY^{2}Z+5X^{2}Z^{2}+3Y^{2}Z^{2}+5Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-6.b.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 30.48.0-6.b.1.2 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-60.p.1.5 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-60.p.1.11 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.1-60.w.1.5 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.48.1-60.w.1.11 | $60$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.288.5-60.jc.1.3 | $60$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 60.480.17-60.id.1.1 | $60$ | $5$ | $5$ | $17$ | $8$ | $1^{16}$ |
| 60.576.17-60.di.1.4 | $60$ | $6$ | $6$ | $17$ | $1$ | $1^{16}$ |
| 60.960.33-60.gk.1.9 | $60$ | $10$ | $10$ | $33$ | $11$ | $1^{32}$ |
| 180.288.5-180.q.1.6 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 180.288.9-180.ch.1.4 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 180.288.9-180.ci.1.7 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |