Invariants
| Level: | $10$ | $\SL_2$-level: | $2$ | ||||
| Index: | $6$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $2^{3}$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 2C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 10.6.0.1 |
Level structure
| $\GL_2(\Z/10\Z)$-generators: | $\begin{bmatrix}5&4\\7&1\end{bmatrix}$, $\begin{bmatrix}7&0\\8&7\end{bmatrix}$ |
| $\GL_2(\Z/10\Z)$-subgroup: | $\GL(2,5)$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 10-isogeny field degree: | $6$ |
| Cyclic 10-torsion field degree: | $24$ |
| Full 10-torsion field degree: | $480$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 6941 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{5}\cdot\frac{x^{6}(60x^{2}+y^{2})^{3}}{x^{8}(20x^{2}-y^{2})^{2}}$ |
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 |
|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $2$ | $2$ | $0$ | $0$ |
| 10.2.0.a.1 | $10$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 10.30.2.c.1 | $10$ | $5$ | $5$ | $2$ |
| 10.36.1.b.1 | $10$ | $6$ | $6$ | $1$ |
| 10.60.3.e.1 | $10$ | $10$ | $10$ | $3$ |
| 20.12.0.e.1 | $20$ | $2$ | $2$ | $0$ |
| 20.12.0.g.1 | $20$ | $2$ | $2$ | $0$ |
| 20.12.0.i.1 | $20$ | $2$ | $2$ | $0$ |
| 20.12.0.k.1 | $20$ | $2$ | $2$ | $0$ |
| 30.18.1.a.1 | $30$ | $3$ | $3$ | $1$ |
| 30.24.0.a.1 | $30$ | $4$ | $4$ | $0$ |
| 40.12.0.n.1 | $40$ | $2$ | $2$ | $0$ |
| 40.12.0.t.1 | $40$ | $2$ | $2$ | $0$ |
| 40.12.0.bc.1 | $40$ | $2$ | $2$ | $0$ |
| 40.12.0.bi.1 | $40$ | $2$ | $2$ | $0$ |
| 60.12.0.m.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.o.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.z.1 | $60$ | $2$ | $2$ | $0$ |
| 60.12.0.bb.1 | $60$ | $2$ | $2$ | $0$ |
| 70.48.2.c.1 | $70$ | $8$ | $8$ | $2$ |
| 70.126.7.a.1 | $70$ | $21$ | $21$ | $7$ |
| 70.168.9.d.1 | $70$ | $28$ | $28$ | $9$ |
| 90.162.10.a.1 | $90$ | $27$ | $27$ | $10$ |
| 110.72.4.a.1 | $110$ | $12$ | $12$ | $4$ |
| 110.330.21.a.1 | $110$ | $55$ | $55$ | $21$ |
| 110.330.21.b.1 | $110$ | $55$ | $55$ | $21$ |
| 120.12.0.bp.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.bv.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.da.1 | $120$ | $2$ | $2$ | $0$ |
| 120.12.0.dg.1 | $120$ | $2$ | $2$ | $0$ |
| 130.84.5.a.1 | $130$ | $14$ | $14$ | $5$ |
| 140.12.0.i.1 | $140$ | $2$ | $2$ | $0$ |
| 140.12.0.j.1 | $140$ | $2$ | $2$ | $0$ |
| 140.12.0.q.1 | $140$ | $2$ | $2$ | $0$ |
| 140.12.0.r.1 | $140$ | $2$ | $2$ | $0$ |
| 170.108.7.a.1 | $170$ | $18$ | $18$ | $7$ |
| 190.120.8.a.1 | $190$ | $20$ | $20$ | $8$ |
| 220.12.0.i.1 | $220$ | $2$ | $2$ | $0$ |
| 220.12.0.j.1 | $220$ | $2$ | $2$ | $0$ |
| 220.12.0.q.1 | $220$ | $2$ | $2$ | $0$ |
| 220.12.0.r.1 | $220$ | $2$ | $2$ | $0$ |
| 230.144.10.a.1 | $230$ | $24$ | $24$ | $10$ |
| 260.12.0.i.1 | $260$ | $2$ | $2$ | $0$ |
| 260.12.0.j.1 | $260$ | $2$ | $2$ | $0$ |
| 260.12.0.q.1 | $260$ | $2$ | $2$ | $0$ |
| 260.12.0.r.1 | $260$ | $2$ | $2$ | $0$ |
| 280.12.0.bc.1 | $280$ | $2$ | $2$ | $0$ |
| 280.12.0.bf.1 | $280$ | $2$ | $2$ | $0$ |
| 280.12.0.ca.1 | $280$ | $2$ | $2$ | $0$ |
| 280.12.0.cd.1 | $280$ | $2$ | $2$ | $0$ |
| 290.180.13.a.1 | $290$ | $30$ | $30$ | $13$ |
| 310.192.14.c.1 | $310$ | $32$ | $32$ | $14$ |