Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $3600$ | ||
| Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{4}\cdot10^{4}\cdot20^{4}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20I5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.144.5.609 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}23&10\\20&59\end{bmatrix}$, $\begin{bmatrix}37&50\\54&7\end{bmatrix}$, $\begin{bmatrix}43&40\\52&51\end{bmatrix}$, $\begin{bmatrix}51&35\\10&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $8$ |
| Cyclic 60-torsion field degree: | $64$ |
| Full 60-torsion field degree: | $15360$ |
Jacobian
| Conductor: | $2^{16}\cdot3^{8}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 80.2.a.a, 360.2.f.c, 900.2.a.b, 3600.2.a.be |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y w + w^{2} - t^{2} $ |
| $=$ | $4 x^{2} - x z + z^{2} - t^{2}$ | |
| $=$ | $3 x^{2} + 3 x z - y^{2} + 2 y w - 3 z^{2} - 2 w^{2} + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 400 x^{8} + 525 x^{6} y^{2} - 920 x^{6} z^{2} + 225 x^{4} y^{4} - 660 x^{4} y^{2} z^{2} + 689 x^{4} z^{4} + \cdots + 16 z^{8} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Maps to other modular curves
$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{3^2}{2^{34}}\cdot\frac{570461009765625xz^{17}-3159422212500000xz^{15}t^{2}+6729935726250000xz^{13}t^{4}-7219443492000000xz^{11}t^{6}+4068132012000000xz^{9}t^{8}-1075913210880000xz^{7}t^{10}+211423030272000xz^{5}t^{12}+171954153062400xz^{3}t^{14}+238560056770560xzt^{16}-117051486328125z^{18}+892462587890625z^{16}t^{2}-2190865750312500z^{14}t^{4}+2318799210750000z^{12}t^{6}-980087655600000z^{10}t^{8}-1990841760000z^{8}t^{10}+67773239040000z^{6}t^{12}-116095862169600z^{4}t^{14}-261385758965760z^{2}t^{16}+3728031612928000w^{18}-22368189677568000w^{16}t^{2}+60387240181760000w^{14}t^{4}-97787815395328000w^{12}t^{6}+106858786324480000w^{10}t^{8}-83906481094656000w^{8}t^{10}+48777084587212800w^{6}t^{12}-20767025869619200w^{4}t^{14}+5655612935372800w^{2}t^{16}-600568391729152t^{18}}{t^{4}w^{2}(w-t)^{5}(w+t)^{5}(5w^{2}-t^{2})}$ |
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 |
|---|---|---|---|---|---|---|
| 20.72.1.u.2 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 60.72.1.cl.2 | $60$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 60.72.1.co.1 | $60$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 60.72.3.rd.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.rh.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.rw.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 60.72.3.zd.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.288.13.py.2 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{4}\cdot2^{2}$ |
| 60.288.13.pz.1 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{4}\cdot2^{2}$ |
| 60.288.13.qa.2 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{4}\cdot2^{2}$ |
| 60.288.13.qb.1 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{4}\cdot2^{2}$ |
| 60.432.29.dzw.1 | $60$ | $3$ | $3$ | $29$ | $4$ | $1^{12}\cdot2^{6}$ |
| 60.576.33.ph.1 | $60$ | $4$ | $4$ | $33$ | $7$ | $1^{14}\cdot2^{7}$ |
| 60.720.37.pv.1 | $60$ | $5$ | $5$ | $37$ | $9$ | $1^{16}\cdot2^{8}$ |
| 120.288.13.jes.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.13.jet.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.13.jgo.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.13.jgp.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.13.jhk.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.13.jhl.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.13.jhm.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.13.jhn.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.13.jji.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.13.jjj.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.13.jjy.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.13.jjz.2 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.17.bbjd.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.bbjh.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cmlb.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cmld.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.csjb.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.csjd.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.ctcx.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.ctdb.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |