Invariants
| Level: | $5$ | $\SL_2$-level: | $5$ | ||||
| Index: | $60$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $0 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $5^{12}$ | Cusp orbits | $1^{2}\cdot2\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 5H0 |
| Sutherland and Zywina (SZ) label: | 5H0-5a |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 5.60.0.1 |
| Sutherland (S) label: | 5Cs.4.1 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 60 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{60}(x^{4}+3x^{3}y-x^{2}y^{2}-3xy^{3}+y^{4})^{3}(x^{8}-4x^{7}y+7x^{6}y^{2}-2x^{5}y^{3}+15x^{4}y^{4}+2x^{3}y^{5}+7x^{2}y^{6}+4xy^{7}+y^{8})^{3}(x^{8}+x^{7}y+7x^{6}y^{2}-7x^{5}y^{3}+7x^{3}y^{5}+7x^{2}y^{6}-xy^{7}+y^{8})^{3}}{y^{5}x^{65}(x^{2}-xy-y^{2})^{5}(x^{4}-2x^{3}y+4x^{2}y^{2}-3xy^{3}+y^{4})^{5}(x^{4}+3x^{3}y+4x^{2}y^{2}+2xy^{3}+y^{4})^{5}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\pm1}(5)$ | $5$ | $5$ | $5$ | $0$ | $0$ |
| 5.12.0.a.2 | $5$ | $5$ | $5$ | $0$ | $0$ |
| $X_{\mathrm{sp}}(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 10.120.5.c.1 | $10$ | $2$ | $2$ | $5$ |
| 10.120.5.e.1 | $10$ | $2$ | $2$ | $5$ |
| 10.180.4.a.1 | $10$ | $3$ | $3$ | $4$ |
| 15.180.10.b.1 | $15$ | $3$ | $3$ | $10$ |
| 15.240.9.a.1 | $15$ | $4$ | $4$ | $9$ |
| 20.120.5.p.1 | $20$ | $2$ | $2$ | $5$ |
| 20.120.5.u.1 | $20$ | $2$ | $2$ | $5$ |
| 20.240.15.bj.1 | $20$ | $4$ | $4$ | $15$ |
| 25.300.12.a.1 | $25$ | $5$ | $5$ | $12$ |
| 25.300.12.b.1 | $25$ | $5$ | $5$ | $12$ |
| 25.300.12.c.1 | $25$ | $5$ | $5$ | $12$ |
| 25.300.12.d.1 | $25$ | $5$ | $5$ | $12$ |
| 25.300.12.e.1 | $25$ | $5$ | $5$ | $12$ |
| 25.300.16.a.1 | $25$ | $5$ | $5$ | $16$ |
| 25.300.16.a.2 | $25$ | $5$ | $5$ | $16$ |
| 30.120.5.e.1 | $30$ | $2$ | $2$ | $5$ |
| 30.120.5.m.1 | $30$ | $2$ | $2$ | $5$ |
| 35.480.29.a.1 | $35$ | $8$ | $8$ | $29$ |
| 35.1260.88.a.1 | $35$ | $21$ | $21$ | $88$ |
| 35.1680.117.a.1 | $35$ | $28$ | $28$ | $117$ |
| 40.120.5.bx.1 | $40$ | $2$ | $2$ | $5$ |
| 40.120.5.cd.1 | $40$ | $2$ | $2$ | $5$ |
| 40.120.5.cu.1 | $40$ | $2$ | $2$ | $5$ |
| 40.120.5.cx.1 | $40$ | $2$ | $2$ | $5$ |
| 45.1620.118.i.1 | $45$ | $27$ | $27$ | $118$ |
| 55.720.49.a.1 | $55$ | $12$ | $12$ | $49$ |
| 55.3300.246.a.1 | $55$ | $55$ | $55$ | $246$ |
| 55.3300.246.b.1 | $55$ | $55$ | $55$ | $246$ |
| 55.3960.295.a.1 | $55$ | $66$ | $66$ | $295$ |
| 60.120.5.v.1 | $60$ | $2$ | $2$ | $5$ |
| 60.120.5.cm.1 | $60$ | $2$ | $2$ | $5$ |
| 65.840.59.a.1 | $65$ | $14$ | $14$ | $59$ |
| 65.4680.355.a.1 | $65$ | $78$ | $78$ | $355$ |
| 65.5460.414.a.1 | $65$ | $91$ | $91$ | $414$ |
| 65.5460.414.b.1 | $65$ | $91$ | $91$ | $414$ |
| 70.120.5.j.1 | $70$ | $2$ | $2$ | $5$ |
| 70.120.5.k.1 | $70$ | $2$ | $2$ | $5$ |
| 110.120.5.e.1 | $110$ | $2$ | $2$ | $5$ |
| 110.120.5.f.1 | $110$ | $2$ | $2$ | $5$ |
| 120.120.5.cv.1 | $120$ | $2$ | $2$ | $5$ |
| 120.120.5.db.1 | $120$ | $2$ | $2$ | $5$ |
| 120.120.5.iq.1 | $120$ | $2$ | $2$ | $5$ |
| 120.120.5.it.1 | $120$ | $2$ | $2$ | $5$ |
| 130.120.5.j.1 | $130$ | $2$ | $2$ | $5$ |
| 130.120.5.k.1 | $130$ | $2$ | $2$ | $5$ |
| 140.120.5.z.1 | $140$ | $2$ | $2$ | $5$ |
| 140.120.5.bc.1 | $140$ | $2$ | $2$ | $5$ |
| 170.120.5.e.1 | $170$ | $2$ | $2$ | $5$ |
| 170.120.5.f.1 | $170$ | $2$ | $2$ | $5$ |
| 190.120.5.j.1 | $190$ | $2$ | $2$ | $5$ |
| 190.120.5.k.1 | $190$ | $2$ | $2$ | $5$ |
| 210.120.5.r.1 | $210$ | $2$ | $2$ | $5$ |
| 210.120.5.u.1 | $210$ | $2$ | $2$ | $5$ |
| 220.120.5.u.1 | $220$ | $2$ | $2$ | $5$ |
| 220.120.5.x.1 | $220$ | $2$ | $2$ | $5$ |
| 230.120.5.e.1 | $230$ | $2$ | $2$ | $5$ |
| 230.120.5.f.1 | $230$ | $2$ | $2$ | $5$ |
| 260.120.5.z.1 | $260$ | $2$ | $2$ | $5$ |
| 260.120.5.bc.1 | $260$ | $2$ | $2$ | $5$ |
| 275.300.12.a.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.b.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.c.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.d.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.e.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.f.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.g.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.h.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.i.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.j.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.k.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.l.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.m.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.n.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.o.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.p.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.q.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.r.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.s.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.t.1 | $275$ | $5$ | $5$ | $12$ |
| 280.120.5.de.1 | $280$ | $2$ | $2$ | $5$ |
| 280.120.5.dh.1 | $280$ | $2$ | $2$ | $5$ |
| 280.120.5.dq.1 | $280$ | $2$ | $2$ | $5$ |
| 280.120.5.dt.1 | $280$ | $2$ | $2$ | $5$ |
| 290.120.5.e.1 | $290$ | $2$ | $2$ | $5$ |
| 290.120.5.f.1 | $290$ | $2$ | $2$ | $5$ |
| 310.120.5.j.1 | $310$ | $2$ | $2$ | $5$ |
| 310.120.5.k.1 | $310$ | $2$ | $2$ | $5$ |
| 330.120.5.m.1 | $330$ | $2$ | $2$ | $5$ |
| 330.120.5.p.1 | $330$ | $2$ | $2$ | $5$ |