Invariants
| Level: | $260$ | $\SL_2$-level: | $20$ | ||||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $0 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot2^{2}\cdot5^{2}\cdot10^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10F0 |
Level structure
| $\GL_2(\Z/260\Z)$-generators: | $\begin{bmatrix}49&142\\28&63\end{bmatrix}$, $\begin{bmatrix}68&193\\137&84\end{bmatrix}$, $\begin{bmatrix}161&258\\118&31\end{bmatrix}$, $\begin{bmatrix}209&70\\230&199\end{bmatrix}$, $\begin{bmatrix}241&70\\102&199\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 10.36.0.a.2 for the level structure with $-I$) |
| Cyclic 260-isogeny field degree: | $28$ |
| Cyclic 260-torsion field degree: | $1344$ |
| Full 260-torsion field degree: | $16773120$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 42 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^5}\cdot\frac{x^{36}(x^{12}+32x^{11}y+416x^{10}y^{2}+2880x^{9}y^{3}+11520x^{8}y^{4}+18432x^{7}y^{5}-65536x^{6}y^{6}-442368x^{5}y^{7}-983040x^{4}y^{8}-655360x^{3}y^{9}+1048576x^{2}y^{10}+2097152xy^{11}+1048576y^{12})^{3}}{y^{5}x^{46}(x+2y)^{5}(x+4y)^{10}(x^{2}+2xy-4y^{2})(x^{2}+12xy+16y^{2})^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 260.144.1-10.a.1.8 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-10.b.1.3 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-20.b.1.10 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-20.d.2.3 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-130.e.1.2 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-20.f.2.1 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-130.f.1.9 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-20.i.1.6 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-20.k.2.4 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-20.m.2.2 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-260.q.1.7 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-260.t.2.1 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-260.u.2.9 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-260.x.1.11 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-260.ba.2.3 | $260$ | $2$ | $2$ | $1$ |
| 260.144.1-260.bb.2.7 | $260$ | $2$ | $2$ | $1$ |
| 260.144.3-20.bg.2.3 | $260$ | $2$ | $2$ | $3$ |
| 260.144.3-20.bi.2.1 | $260$ | $2$ | $2$ | $3$ |
| 260.144.3-20.bk.2.4 | $260$ | $2$ | $2$ | $3$ |
| 260.144.3-20.bm.2.2 | $260$ | $2$ | $2$ | $3$ |
| 260.144.3-260.cm.2.5 | $260$ | $2$ | $2$ | $3$ |
| 260.144.3-260.cn.2.13 | $260$ | $2$ | $2$ | $3$ |
| 260.144.3-260.cu.2.7 | $260$ | $2$ | $2$ | $3$ |
| 260.144.3-260.cv.2.15 | $260$ | $2$ | $2$ | $3$ |
| 260.360.4-10.a.1.3 | $260$ | $5$ | $5$ | $4$ |