Invariants
| Level: | $10$ | $\SL_2$-level: | $10$ | ||||
| Index: | $20$ | $\PSL_2$-index: | $20$ | ||||
| Genus: | $0 = 1 + \frac{ 20 }{12} - \frac{ 4 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (none of which are rational) | Cusp widths | $10^{2}$ | Cusp orbits | $2$ | ||
| Elliptic points: | $4$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3$) |
Other labels
| Cummins and Pauli (CP) label: | 10D0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 10.20.0.2 |
Level structure
| $\GL_2(\Z/10\Z)$-generators: | $\begin{bmatrix}0&3\\7&0\end{bmatrix}$, $\begin{bmatrix}3&3\\1&2\end{bmatrix}$ |
| $\GL_2(\Z/10\Z)$-subgroup: | $C_{12}.D_6$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 10-isogeny field degree: | $18$ |
| Cyclic 10-torsion field degree: | $72$ |
| Full 10-torsion field degree: | $144$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 20 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -5^4\,\frac{(x-3y)^{3}(x+y)^{20}(x+5y)^{3}(x^{2}+10xy-55y^{2})(19x^{4}-60x^{3}y+170x^{2}y^{2}-300xy^{3}+475y^{4})^{3}}{(x+y)^{20}(x^{2}-10xy+5y^{2})^{10}}$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 10.40.1.b.1 | $10$ | $2$ | $2$ | $1$ |
| 10.40.1.c.1 | $10$ | $2$ | $2$ | $1$ |
| 10.60.2.d.1 | $10$ | $3$ | $3$ | $2$ |
| 10.60.2.f.1 | $10$ | $3$ | $3$ | $2$ |
| 20.40.1.f.1 | $20$ | $2$ | $2$ | $1$ |
| 20.40.1.i.1 | $20$ | $2$ | $2$ | $1$ |
| 20.80.3.b.1 | $20$ | $4$ | $4$ | $3$ |
| 30.40.1.e.1 | $30$ | $2$ | $2$ | $1$ |
| 30.40.1.i.1 | $30$ | $2$ | $2$ | $1$ |
| 30.60.2.h.1 | $30$ | $3$ | $3$ | $2$ |
| 30.80.5.f.1 | $30$ | $4$ | $4$ | $5$ |
| 40.40.1.o.1 | $40$ | $2$ | $2$ | $1$ |
| 40.40.1.x.1 | $40$ | $2$ | $2$ | $1$ |
| 40.40.1.ba.1 | $40$ | $2$ | $2$ | $1$ |
| 40.40.1.bj.1 | $40$ | $2$ | $2$ | $1$ |
| 50.100.4.b.1 | $50$ | $5$ | $5$ | $4$ |
| 50.500.32.a.1 | $50$ | $25$ | $25$ | $32$ |
| 60.40.1.o.1 | $60$ | $2$ | $2$ | $1$ |
| 60.40.1.ba.1 | $60$ | $2$ | $2$ | $1$ |
| 70.40.1.i.1 | $70$ | $2$ | $2$ | $1$ |
| 70.40.1.k.1 | $70$ | $2$ | $2$ | $1$ |
| 70.160.11.g.1 | $70$ | $8$ | $8$ | $11$ |
| 70.420.28.c.1 | $70$ | $21$ | $21$ | $28$ |
| 70.560.39.g.1 | $70$ | $28$ | $28$ | $39$ |
| 110.40.1.i.1 | $110$ | $2$ | $2$ | $1$ |
| 110.40.1.k.1 | $110$ | $2$ | $2$ | $1$ |
| 110.240.19.g.1 | $110$ | $12$ | $12$ | $19$ |
| 120.40.1.cb.1 | $120$ | $2$ | $2$ | $1$ |
| 120.40.1.ch.1 | $120$ | $2$ | $2$ | $1$ |
| 120.40.1.du.1 | $120$ | $2$ | $2$ | $1$ |
| 120.40.1.ed.1 | $120$ | $2$ | $2$ | $1$ |
| 130.40.1.i.1 | $130$ | $2$ | $2$ | $1$ |
| 130.40.1.k.1 | $130$ | $2$ | $2$ | $1$ |
| 130.280.19.e.1 | $130$ | $14$ | $14$ | $19$ |
| 140.40.1.ba.1 | $140$ | $2$ | $2$ | $1$ |
| 140.40.1.bg.1 | $140$ | $2$ | $2$ | $1$ |
| 170.40.1.i.1 | $170$ | $2$ | $2$ | $1$ |
| 170.40.1.k.1 | $170$ | $2$ | $2$ | $1$ |
| 190.40.1.i.1 | $190$ | $2$ | $2$ | $1$ |
| 190.40.1.k.1 | $190$ | $2$ | $2$ | $1$ |
| 210.40.1.ba.1 | $210$ | $2$ | $2$ | $1$ |
| 210.40.1.bg.1 | $210$ | $2$ | $2$ | $1$ |
| 220.40.1.ba.1 | $220$ | $2$ | $2$ | $1$ |
| 220.40.1.bg.1 | $220$ | $2$ | $2$ | $1$ |
| 230.40.1.i.1 | $230$ | $2$ | $2$ | $1$ |
| 230.40.1.k.1 | $230$ | $2$ | $2$ | $1$ |
| 260.40.1.ba.1 | $260$ | $2$ | $2$ | $1$ |
| 260.40.1.bg.1 | $260$ | $2$ | $2$ | $1$ |
| 280.40.1.ea.1 | $280$ | $2$ | $2$ | $1$ |
| 280.40.1.ed.1 | $280$ | $2$ | $2$ | $1$ |
| 280.40.1.ey.1 | $280$ | $2$ | $2$ | $1$ |
| 280.40.1.fb.1 | $280$ | $2$ | $2$ | $1$ |
| 290.40.1.i.1 | $290$ | $2$ | $2$ | $1$ |
| 290.40.1.k.1 | $290$ | $2$ | $2$ | $1$ |
| 310.40.1.i.1 | $310$ | $2$ | $2$ | $1$ |
| 310.40.1.k.1 | $310$ | $2$ | $2$ | $1$ |
| 330.40.1.ba.1 | $330$ | $2$ | $2$ | $1$ |
| 330.40.1.bg.1 | $330$ | $2$ | $2$ | $1$ |