Invariants
| Level: | $25$ | $\SL_2$-level: | $25$ | ||||
| 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 | $1^{10}\cdot25^{2}$ | 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: | 25B0 |
| Sutherland and Zywina (SZ) label: | 25B0-25b |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 25.60.0.2 |
Level structure
| $\GL_2(\Z/25\Z)$-generators: | $\begin{bmatrix}4&7\\0&19\end{bmatrix}$, $\begin{bmatrix}18&19\\0&16\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 25.120.0-25.a.2.1, 25.120.0-25.a.2.2, 50.120.0-25.a.2.1, 50.120.0-25.a.2.2, 75.120.0-25.a.2.1, 75.120.0-25.a.2.2, 100.120.0-25.a.2.1, 100.120.0-25.a.2.2, 100.120.0-25.a.2.3, 100.120.0-25.a.2.4, 150.120.0-25.a.2.1, 150.120.0-25.a.2.2, 175.120.0-25.a.2.1, 175.120.0-25.a.2.2, 200.120.0-25.a.2.1, 200.120.0-25.a.2.2, 200.120.0-25.a.2.3, 200.120.0-25.a.2.4, 200.120.0-25.a.2.5, 200.120.0-25.a.2.6, 200.120.0-25.a.2.7, 200.120.0-25.a.2.8, 275.120.0-25.a.2.1, 275.120.0-25.a.2.2, 300.120.0-25.a.2.1, 300.120.0-25.a.2.2, 300.120.0-25.a.2.3, 300.120.0-25.a.2.4, 325.120.0-25.a.2.1, 325.120.0-25.a.2.2 |
| Cyclic 25-isogeny field degree: | $1$ |
| Cyclic 25-torsion field degree: | $20$ |
| Full 25-torsion field degree: | $5000$ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(x-y)^{60}(1844144x^{20}-13905320x^{19}y+48979760x^{18}y^{2}-105184440x^{17}y^{3}+148393605x^{16}y^{4}-131615424x^{15}y^{5}+44718840x^{14}y^{6}+64400580x^{13}y^{7}-130561620x^{12}y^{8}+128577640x^{11}y^{9}-83857386x^{10}y^{10}+36648140x^{9}y^{11}-8423370x^{8}y^{12}-1656420x^{7}y^{13}+2723340x^{6}y^{14}-1521924x^{5}y^{15}+567105x^{4}y^{16}-152940x^{3}y^{17}+29510x^{2}y^{18}-3820xy^{19}+269y^{20})^{3}}{(x-y)^{60}(x+y)(3x-2y)(x^{2}-3xy+y^{2})^{25}(11x^{4}-31x^{3}y+41x^{2}y^{2}-31xy^{3}+11y^{4})(41x^{4}-51x^{3}y+26x^{2}y^{2}-6xy^{3}+y^{4})}$ |
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 |
|---|---|---|---|---|---|
| 5.12.0.a.2 | $5$ | $5$ | $5$ | $0$ | $0$ |
| $X_0(25)$ | $25$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 25.300.12.g.2 | $25$ | $5$ | $5$ | $12$ |
| 25.300.12.h.1 | $25$ | $5$ | $5$ | $12$ |
| 25.300.12.i.1 | $25$ | $5$ | $5$ | $12$ |
| 25.300.12.j.2 | $25$ | $5$ | $5$ | $12$ |
| 25.300.12.k.2 | $25$ | $5$ | $5$ | $12$ |
| 25.300.16.a.2 | $25$ | $5$ | $5$ | $16$ |
| 50.120.5.a.2 | $50$ | $2$ | $2$ | $5$ |
| 50.120.5.b.2 | $50$ | $2$ | $2$ | $5$ |
| 50.180.4.a.1 | $50$ | $3$ | $3$ | $4$ |
| 75.180.10.a.2 | $75$ | $3$ | $3$ | $10$ |
| 75.240.9.a.2 | $75$ | $4$ | $4$ | $9$ |
| 100.120.5.b.2 | $100$ | $2$ | $2$ | $5$ |
| 100.120.5.e.2 | $100$ | $2$ | $2$ | $5$ |
| 100.240.15.i.2 | $100$ | $4$ | $4$ | $15$ |
| 125.300.16.a.2 | $125$ | $5$ | $5$ | $16$ |
| 150.120.5.b.2 | $150$ | $2$ | $2$ | $5$ |
| 150.120.5.e.2 | $150$ | $2$ | $2$ | $5$ |
| 200.120.5.b.2 | $200$ | $2$ | $2$ | $5$ |
| 200.120.5.h.2 | $200$ | $2$ | $2$ | $5$ |
| 200.120.5.n.2 | $200$ | $2$ | $2$ | $5$ |
| 200.120.5.t.2 | $200$ | $2$ | $2$ | $5$ |
| 275.300.12.u.2 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.v.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.w.2 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.x.2 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.y.2 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.z.2 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.ba.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.bb.2 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.bc.2 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.bd.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.be.2 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.bf.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.bg.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.bh.2 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.bi.2 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.bj.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.bk.1 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.bl.2 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.bm.2 | $275$ | $5$ | $5$ | $12$ |
| 275.300.12.bn.2 | $275$ | $5$ | $5$ | $12$ |
| 300.120.5.e.2 | $300$ | $2$ | $2$ | $5$ |
| 300.120.5.n.2 | $300$ | $2$ | $2$ | $5$ |