Invariants
| Level: | $60$ | $\SL_2$-level: | $15$ | Newform level: | $3600$ | ||
| Index: | $90$ | $\PSL_2$-index: | $90$ | ||||
| Genus: | $4 = 1 + \frac{ 90 }{12} - \frac{ 6 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $15^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $6$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $4$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 15D4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.90.4.16 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}5&51\\3&20\end{bmatrix}$, $\begin{bmatrix}10&41\\43&10\end{bmatrix}$, $\begin{bmatrix}16&55\\25&8\end{bmatrix}$, $\begin{bmatrix}22&45\\45&44\end{bmatrix}$, $\begin{bmatrix}56&25\\5&2\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $768$ |
| Full 60-torsion field degree: | $24576$ |
Jacobian
| Conductor: | $2^{12}\cdot3^{8}\cdot5^{8}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{2}\cdot2$ |
| Newforms: | 225.2.a.c, 3600.2.a.b, 3600.2.a.bs |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 75 x^{2} + 11 y^{2} + 6 y z - z^{2} $ |
| $=$ | $y^{3} + y^{2} z - y z^{2} - w^{3}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} y^{2} + 75 x^{2} y^{4} + 6 x^{2} y z^{3} + 125 y^{6} + 50 y^{3} z^{3} + z^{6} $ |
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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{5}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 90 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{65859376y^{2}z^{13}-117078112y^{2}z^{10}w^{3}+63665664y^{2}z^{7}w^{6}-10746880y^{2}z^{4}w^{9}+286720y^{2}zw^{12}-40703124yz^{14}+90562512yz^{11}w^{3}-66346848yz^{8}w^{6}+17481216yz^{5}w^{9}-1136640yz^{2}w^{12}-z^{15}-40703139z^{12}w^{3}+65406200z^{9}w^{6}-30221888z^{6}w^{9}+3709440z^{3}w^{12}-32768w^{15}}{w^{15}}$ |
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 |
|---|---|---|---|---|---|---|
| 15.45.1.a.1 | $15$ | $2$ | $2$ | $1$ | $1$ | $1\cdot2$ |
| 60.30.0.a.1 | $60$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.180.9.bc.1 | $60$ | $2$ | $2$ | $9$ | $4$ | $1^{5}$ |
| 60.180.9.bd.1 | $60$ | $2$ | $2$ | $9$ | $7$ | $1^{5}$ |
| 60.180.9.bi.1 | $60$ | $2$ | $2$ | $9$ | $5$ | $1^{5}$ |
| 60.180.9.bj.1 | $60$ | $2$ | $2$ | $9$ | $5$ | $1^{5}$ |
| 60.180.9.ca.1 | $60$ | $2$ | $2$ | $9$ | $5$ | $1^{5}$ |
| 60.180.9.cb.1 | $60$ | $2$ | $2$ | $9$ | $6$ | $1^{5}$ |
| 60.180.9.cg.1 | $60$ | $2$ | $2$ | $9$ | $6$ | $1^{5}$ |
| 60.180.9.ch.1 | $60$ | $2$ | $2$ | $9$ | $4$ | $1^{5}$ |
| 60.180.11.u.1 | $60$ | $2$ | $2$ | $11$ | $7$ | $1^{7}$ |
| 60.180.11.v.1 | $60$ | $2$ | $2$ | $11$ | $6$ | $1^{7}$ |
| 60.180.11.ba.1 | $60$ | $2$ | $2$ | $11$ | $6$ | $1^{7}$ |
| 60.180.11.bb.1 | $60$ | $2$ | $2$ | $11$ | $7$ | $1^{7}$ |
| 60.180.11.bs.1 | $60$ | $2$ | $2$ | $11$ | $5$ | $1^{7}$ |
| 60.180.11.bt.1 | $60$ | $2$ | $2$ | $11$ | $4$ | $1^{7}$ |
| 60.180.11.by.1 | $60$ | $2$ | $2$ | $11$ | $7$ | $1^{7}$ |
| 60.180.11.bz.1 | $60$ | $2$ | $2$ | $11$ | $8$ | $1^{7}$ |
| 60.180.13.bi.1 | $60$ | $2$ | $2$ | $13$ | $5$ | $1^{9}$ |
| 60.180.13.bk.1 | $60$ | $2$ | $2$ | $13$ | $5$ | $1^{9}$ |
| 60.180.13.ez.1 | $60$ | $2$ | $2$ | $13$ | $6$ | $1^{9}$ |
| 60.180.13.fb.1 | $60$ | $2$ | $2$ | $13$ | $6$ | $1^{9}$ |
| 60.180.13.fl.1 | $60$ | $2$ | $2$ | $13$ | $6$ | $1^{9}$ |
| 60.180.13.fn.1 | $60$ | $2$ | $2$ | $13$ | $6$ | $1^{9}$ |
| 60.180.13.fr.1 | $60$ | $2$ | $2$ | $13$ | $6$ | $1^{9}$ |
| 60.180.13.ft.1 | $60$ | $2$ | $2$ | $13$ | $6$ | $1^{9}$ |
| 60.270.16.b.1 | $60$ | $3$ | $3$ | $16$ | $10$ | $1^{10}\cdot2$ |
| 60.360.25.cgd.1 | $60$ | $4$ | $4$ | $25$ | $16$ | $1^{21}$ |
| 120.180.9.dm.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.180.9.dp.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.180.9.ek.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.180.9.en.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.180.9.gs.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.180.9.gv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.180.9.hq.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.180.9.ht.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.180.11.da.1 | $120$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 120.180.11.dd.1 | $120$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 120.180.11.dy.1 | $120$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 120.180.11.eb.1 | $120$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 120.180.11.gs.1 | $120$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 120.180.11.gv.1 | $120$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 120.180.11.hq.1 | $120$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 120.180.11.ht.1 | $120$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 120.180.13.em.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.180.13.es.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.180.13.rt.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.180.13.sc.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.180.13.tw.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.180.13.uc.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.180.13.uu.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.180.13.va.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 180.270.18.f.1 | $180$ | $3$ | $3$ | $18$ | $?$ | not computed |
| 300.450.24.a.1 | $300$ | $5$ | $5$ | $24$ | $?$ | not computed |