Invariants
| Level: | $10$ | $\SL_2$-level: | $10$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot10$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $4$ 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: | 10B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 10.12.0.2 |
Level structure
| $\GL_2(\Z/10\Z)$-generators: | $\begin{bmatrix}3&7\\2&1\end{bmatrix}$, $\begin{bmatrix}6&5\\3&7\end{bmatrix}$ |
| $\GL_2(\Z/10\Z)$-subgroup: | $C_{15}:C_4^2$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 10-isogeny field degree: | $3$ |
| Cyclic 10-torsion field degree: | $12$ |
| Full 10-torsion field degree: | $240$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 8 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{12}(5x^{4}+160x^{2}y^{2}+256y^{4})^{3}}{y^{2}x^{22}}$ |
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 |
|---|---|---|---|---|---|
| $X_0(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 10.24.1.a.1 | $10$ | $2$ | $2$ | $1$ |
| 10.24.1.b.2 | $10$ | $2$ | $2$ | $1$ |
| 10.36.0.b.2 | $10$ | $3$ | $3$ | $0$ |
| 10.60.2.c.1 | $10$ | $5$ | $5$ | $2$ |
| 20.24.1.c.1 | $20$ | $2$ | $2$ | $1$ |
| 20.24.1.f.1 | $20$ | $2$ | $2$ | $1$ |
| 20.48.1.b.1 | $20$ | $4$ | $4$ | $1$ |
| 30.24.1.g.1 | $30$ | $2$ | $2$ | $1$ |
| 30.24.1.j.1 | $30$ | $2$ | $2$ | $1$ |
| 30.36.0.f.2 | $30$ | $3$ | $3$ | $0$ |
| 30.48.3.f.1 | $30$ | $4$ | $4$ | $3$ |
| 40.24.1.by.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.ch.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.ck.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.ct.1 | $40$ | $2$ | $2$ | $1$ |
| 50.60.2.a.2 | $50$ | $5$ | $5$ | $2$ |
| 60.24.1.u.1 | $60$ | $2$ | $2$ | $1$ |
| 60.24.1.bh.1 | $60$ | $2$ | $2$ | $1$ |
| 70.24.1.e.1 | $70$ | $2$ | $2$ | $1$ |
| 70.24.1.f.2 | $70$ | $2$ | $2$ | $1$ |
| 70.96.7.f.1 | $70$ | $8$ | $8$ | $7$ |
| 70.252.14.b.1 | $70$ | $21$ | $21$ | $14$ |
| 70.336.21.f.1 | $70$ | $28$ | $28$ | $21$ |
| 90.324.22.d.2 | $90$ | $27$ | $27$ | $22$ |
| 110.24.1.e.1 | $110$ | $2$ | $2$ | $1$ |
| 110.24.1.f.2 | $110$ | $2$ | $2$ | $1$ |
| 110.144.11.f.2 | $110$ | $12$ | $12$ | $11$ |
| 120.24.1.hh.2 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.hn.2 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.kb.2 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.kh.2 | $120$ | $2$ | $2$ | $1$ |
| 130.24.1.e.2 | $130$ | $2$ | $2$ | $1$ |
| 130.24.1.f.1 | $130$ | $2$ | $2$ | $1$ |
| 130.168.11.d.1 | $130$ | $14$ | $14$ | $11$ |
| 140.24.1.o.2 | $140$ | $2$ | $2$ | $1$ |
| 140.24.1.r.2 | $140$ | $2$ | $2$ | $1$ |
| 170.24.1.e.1 | $170$ | $2$ | $2$ | $1$ |
| 170.24.1.f.1 | $170$ | $2$ | $2$ | $1$ |
| 170.216.15.c.2 | $170$ | $18$ | $18$ | $15$ |
| 190.24.1.e.1 | $190$ | $2$ | $2$ | $1$ |
| 190.24.1.f.2 | $190$ | $2$ | $2$ | $1$ |
| 190.240.19.f.1 | $190$ | $20$ | $20$ | $19$ |
| 210.24.1.ba.2 | $210$ | $2$ | $2$ | $1$ |
| 210.24.1.bd.1 | $210$ | $2$ | $2$ | $1$ |
| 220.24.1.o.1 | $220$ | $2$ | $2$ | $1$ |
| 220.24.1.r.2 | $220$ | $2$ | $2$ | $1$ |
| 230.24.1.e.1 | $230$ | $2$ | $2$ | $1$ |
| 230.24.1.f.2 | $230$ | $2$ | $2$ | $1$ |
| 230.288.23.f.2 | $230$ | $24$ | $24$ | $23$ |
| 260.24.1.o.2 | $260$ | $2$ | $2$ | $1$ |
| 260.24.1.r.2 | $260$ | $2$ | $2$ | $1$ |
| 280.24.1.lk.2 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.ln.2 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.lw.2 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.lz.2 | $280$ | $2$ | $2$ | $1$ |
| 290.24.1.e.1 | $290$ | $2$ | $2$ | $1$ |
| 290.24.1.f.1 | $290$ | $2$ | $2$ | $1$ |
| 310.24.1.e.1 | $310$ | $2$ | $2$ | $1$ |
| 310.24.1.f.2 | $310$ | $2$ | $2$ | $1$ |
| 330.24.1.ba.1 | $330$ | $2$ | $2$ | $1$ |
| 330.24.1.bd.2 | $330$ | $2$ | $2$ | $1$ |