Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $48$ | ||
| 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: | 60.96.1.470 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}25&24\\33&7\end{bmatrix}$, $\begin{bmatrix}35&6\\6&13\end{bmatrix}$, $\begin{bmatrix}35&28\\18&55\end{bmatrix}$, $\begin{bmatrix}43&4\\42&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.48.1.bq.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: | $2^{4}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 48.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 5 x y - 3 y^{2} + 2 y z - 2 z^{2} $ |
| $=$ | $45 x^{2} - 20 x y - 5 y^{2} + 20 y z - 20 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} - 24 x^{3} z - 5 x^{2} y^{2} - x^{2} z^{2} + 6 x z^{3} - z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{5448960000000xz^{11}+149328000000xz^{9}w^{2}+6916320000xz^{7}w^{4}+69624000xz^{5}w^{6}+349200xz^{3}w^{8}-528588800000y^{2}z^{10}-56307200000y^{2}z^{8}w^{2}-4149152000y^{2}z^{6}w^{4}-171784000y^{2}z^{4}w^{6}-3889840y^{2}z^{2}w^{8}-37324y^{2}w^{10}+740915200000yz^{11}+206780800000yz^{9}w^{2}+10783648000yz^{7}w^{4}+342440000yz^{5}w^{6}+4598960yz^{3}w^{8}+18656yzw^{10}-3623795200000z^{12}-67065600000z^{10}w^{2}-5162208000z^{8}w^{4}-35296000z^{6}w^{6}-602160z^{4}w^{8}+744z^{2}w^{10}-w^{12}}{w^{4}(69120000xz^{7}+8172000xz^{5}w^{2}+208800xz^{3}w^{4}+24128000y^{2}z^{6}+3368000y^{2}z^{4}w^{2}+121940y^{2}z^{2}w^{4}+729y^{2}w^{6}-44992000yz^{7}-1348000yz^{5}w^{2}+328040yz^{3}w^{4}+11664yzw^{6}-24128000z^{8}-2928000z^{6}w^{2}-78540z^{4}w^{4}-64z^{2}w^{6})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.48.1.bq.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{3}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3z$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}-5X^{2}Y^{2}-24X^{3}Z-X^{2}Z^{2}+6XZ^{3}-Z^{4} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
| 20.12.0.k.1 | $20$ | $8$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.1-12.l.1.10 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.48.0-30.a.1.2 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-30.a.1.8 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-60.p.1.5 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-60.p.1.15 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.1-12.l.1.4 | $60$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.192.3-60.bs.1.8 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.192.3-60.bt.1.6 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.192.3-60.bu.1.8 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.192.3-60.bv.1.4 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.288.5-60.ka.1.1 | $60$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 60.480.17-60.mm.1.12 | $60$ | $5$ | $5$ | $17$ | $5$ | $1^{16}$ |
| 60.576.17-60.ic.1.20 | $60$ | $6$ | $6$ | $17$ | $2$ | $1^{16}$ |
| 60.960.33-60.op.1.23 | $60$ | $10$ | $10$ | $33$ | $8$ | $1^{32}$ |
| 120.192.3-120.vo.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.vp.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.vw.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.vx.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.we.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.wf.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.wg.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.wh.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.wq.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.wr.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.wu.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.wv.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.5-120.by.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ca.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ho.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.hq.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ui.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.uk.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.vo.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.vq.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.288.5-180.bq.1.1 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 180.288.9-180.dy.1.10 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 180.288.9-180.eu.1.10 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |