Invariants
| Level: | $5$ | $\SL_2$-level: | $5$ | ||||
| Index: | $10$ | $\PSL_2$-index: | $10$ | ||||
| Genus: | $0 = 1 + \frac{ 10 }{12} - \frac{ 2 }{4} - \frac{ 1 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (none of which are rational) | Cusp widths | $5^{2}$ | Cusp orbits | $2$ | ||
| Elliptic points: | $2$ of order $2$ and $1$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-7,-8,-12,-27,-28,-43,-67,-163$) |
Other labels
| Cummins and Pauli (CP) label: | 5C0 |
| Sutherland and Zywina (SZ) label: | 5C0-5a |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 5.10.0.1 |
| Sutherland (S) label: | 5Nn |
Level structure
| $\GL_2(\Z/5\Z)$-generators: | $\begin{bmatrix}3&4\\1&2\end{bmatrix}$, $\begin{bmatrix}4&4\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/5\Z)$-subgroup: | $C_{24}:C_2$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 5-isogeny field degree: | $6$ |
| Cyclic 5-torsion field degree: | $24$ |
| Full 5-torsion field degree: | $48$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 63 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 10 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\cdot5^3\,\frac{(x+y)^{3}(2x+y)^{11}(6x^{2}-3xy+y^{2})^{3}}{(2x+y)^{10}(4x^{2}-2xy-y^{2})^{5}}$ |
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 |
|---|---|---|---|---|---|
| $X(1)$ | $1$ | $10$ | $10$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| $X_{\mathrm{ns}}(5)$ | $5$ | $2$ | $2$ | $0$ |
| 5.20.0.b.1 | $5$ | $2$ | $2$ | $0$ |
| 5.30.0.b.1 | $5$ | $3$ | $3$ | $0$ |
| 10.20.0.a.1 | $10$ | $2$ | $2$ | $0$ |
| $X_{\mathrm{ns}}^+(10)$ | $10$ | $2$ | $2$ | $0$ |
| 10.20.1.a.1 | $10$ | $2$ | $2$ | $1$ |
| 10.20.1.b.1 | $10$ | $2$ | $2$ | $1$ |
| 10.30.1.a.1 | $10$ | $3$ | $3$ | $1$ |
| 15.20.0.a.1 | $15$ | $2$ | $2$ | $0$ |
| 15.20.0.b.1 | $15$ | $2$ | $2$ | $0$ |
| 15.30.1.a.1 | $15$ | $3$ | $3$ | $1$ |
| 15.40.2.a.1 | $15$ | $4$ | $4$ | $2$ |
| 20.20.0.a.1 | $20$ | $2$ | $2$ | $0$ |
| 20.20.0.b.1 | $20$ | $2$ | $2$ | $0$ |
| 20.20.0.c.1 | $20$ | $2$ | $2$ | $0$ |
| 20.20.0.d.1 | $20$ | $2$ | $2$ | $0$ |
| 20.20.1.a.1 | $20$ | $2$ | $2$ | $1$ |
| 20.20.1.b.1 | $20$ | $2$ | $2$ | $1$ |
| 20.40.2.e.1 | $20$ | $4$ | $4$ | $2$ |
| 25.50.2.a.1 | $25$ | $5$ | $5$ | $2$ |
| $X_{\mathrm{ns}}^+(25)$ | $25$ | $25$ | $25$ | $14$ |
| 30.20.0.a.1 | $30$ | $2$ | $2$ | $0$ |
| 30.20.0.b.1 | $30$ | $2$ | $2$ | $0$ |
| 30.20.1.a.1 | $30$ | $2$ | $2$ | $1$ |
| 30.20.1.b.1 | $30$ | $2$ | $2$ | $1$ |
| 35.20.0.a.1 | $35$ | $2$ | $2$ | $0$ |
| 35.20.0.b.1 | $35$ | $2$ | $2$ | $0$ |
| 35.80.5.a.1 | $35$ | $8$ | $8$ | $5$ |
| 35.210.13.a.1 | $35$ | $21$ | $21$ | $13$ |
| 35.280.18.a.1 | $35$ | $28$ | $28$ | $18$ |
| 40.20.0.a.1 | $40$ | $2$ | $2$ | $0$ |
| 40.20.0.b.1 | $40$ | $2$ | $2$ | $0$ |
| 40.20.0.c.1 | $40$ | $2$ | $2$ | $0$ |
| 40.20.0.d.1 | $40$ | $2$ | $2$ | $0$ |
| 40.20.0.e.1 | $40$ | $2$ | $2$ | $0$ |
| 40.20.0.f.1 | $40$ | $2$ | $2$ | $0$ |
| 40.20.0.g.1 | $40$ | $2$ | $2$ | $0$ |
| 40.20.0.h.1 | $40$ | $2$ | $2$ | $0$ |
| 40.20.1.a.1 | $40$ | $2$ | $2$ | $1$ |
| 40.20.1.b.1 | $40$ | $2$ | $2$ | $1$ |
| 40.20.1.c.1 | $40$ | $2$ | $2$ | $1$ |
| 40.20.1.d.1 | $40$ | $2$ | $2$ | $1$ |
| 45.270.18.a.1 | $45$ | $27$ | $27$ | $18$ |
| 55.20.0.a.1 | $55$ | $2$ | $2$ | $0$ |
| 55.20.0.b.1 | $55$ | $2$ | $2$ | $0$ |
| 55.120.9.a.1 | $55$ | $12$ | $12$ | $9$ |
| 55.550.38.a.1 | $55$ | $55$ | $55$ | $38$ |
| 55.550.39.a.1 | $55$ | $55$ | $55$ | $39$ |
| 55.660.47.a.1 | $55$ | $66$ | $66$ | $47$ |
| 60.20.0.a.1 | $60$ | $2$ | $2$ | $0$ |
| 60.20.0.b.1 | $60$ | $2$ | $2$ | $0$ |
| 60.20.0.c.1 | $60$ | $2$ | $2$ | $0$ |
| 60.20.0.d.1 | $60$ | $2$ | $2$ | $0$ |
| 60.20.1.a.1 | $60$ | $2$ | $2$ | $1$ |
| 60.20.1.b.1 | $60$ | $2$ | $2$ | $1$ |
| 65.20.0.a.1 | $65$ | $2$ | $2$ | $0$ |
| 65.20.0.b.1 | $65$ | $2$ | $2$ | $0$ |
| 65.140.9.a.1 | $65$ | $14$ | $14$ | $9$ |
| 65.780.57.a.1 | $65$ | $78$ | $78$ | $57$ |
| 65.910.66.a.1 | $65$ | $91$ | $91$ | $66$ |
| 65.910.67.a.1 | $65$ | $91$ | $91$ | $67$ |
| 70.20.0.a.1 | $70$ | $2$ | $2$ | $0$ |
| 70.20.0.b.1 | $70$ | $2$ | $2$ | $0$ |
| 70.20.1.a.1 | $70$ | $2$ | $2$ | $1$ |
| 70.20.1.b.1 | $70$ | $2$ | $2$ | $1$ |
| 85.20.0.a.1 | $85$ | $2$ | $2$ | $0$ |
| 85.20.0.b.1 | $85$ | $2$ | $2$ | $0$ |
| 85.180.13.c.1 | $85$ | $18$ | $18$ | $13$ |
| 95.20.0.a.1 | $95$ | $2$ | $2$ | $0$ |
| 95.20.0.b.1 | $95$ | $2$ | $2$ | $0$ |
| 95.200.15.a.1 | $95$ | $20$ | $20$ | $15$ |
| 105.20.0.a.1 | $105$ | $2$ | $2$ | $0$ |
| 105.20.0.b.1 | $105$ | $2$ | $2$ | $0$ |
| 110.20.0.a.1 | $110$ | $2$ | $2$ | $0$ |
| 110.20.0.b.1 | $110$ | $2$ | $2$ | $0$ |
| 110.20.1.a.1 | $110$ | $2$ | $2$ | $1$ |
| 110.20.1.b.1 | $110$ | $2$ | $2$ | $1$ |
| 115.20.0.a.1 | $115$ | $2$ | $2$ | $0$ |
| 115.20.0.b.1 | $115$ | $2$ | $2$ | $0$ |
| 115.240.19.a.1 | $115$ | $24$ | $24$ | $19$ |
| 120.20.0.a.1 | $120$ | $2$ | $2$ | $0$ |
| 120.20.0.b.1 | $120$ | $2$ | $2$ | $0$ |
| 120.20.0.c.1 | $120$ | $2$ | $2$ | $0$ |
| 120.20.0.d.1 | $120$ | $2$ | $2$ | $0$ |
| 120.20.0.e.1 | $120$ | $2$ | $2$ | $0$ |
| 120.20.0.f.1 | $120$ | $2$ | $2$ | $0$ |
| 120.20.0.g.1 | $120$ | $2$ | $2$ | $0$ |
| 120.20.0.h.1 | $120$ | $2$ | $2$ | $0$ |
| 120.20.1.a.1 | $120$ | $2$ | $2$ | $1$ |
| 120.20.1.b.1 | $120$ | $2$ | $2$ | $1$ |
| 120.20.1.c.1 | $120$ | $2$ | $2$ | $1$ |
| 120.20.1.d.1 | $120$ | $2$ | $2$ | $1$ |
| 130.20.0.a.1 | $130$ | $2$ | $2$ | $0$ |
| 130.20.0.b.1 | $130$ | $2$ | $2$ | $0$ |
| 130.20.1.a.1 | $130$ | $2$ | $2$ | $1$ |
| 130.20.1.b.1 | $130$ | $2$ | $2$ | $1$ |
| 140.20.0.a.1 | $140$ | $2$ | $2$ | $0$ |
| 140.20.0.b.1 | $140$ | $2$ | $2$ | $0$ |
| 140.20.0.c.1 | $140$ | $2$ | $2$ | $0$ |
| 140.20.0.d.1 | $140$ | $2$ | $2$ | $0$ |
| 140.20.1.a.1 | $140$ | $2$ | $2$ | $1$ |
| 140.20.1.b.1 | $140$ | $2$ | $2$ | $1$ |
| 145.20.0.a.1 | $145$ | $2$ | $2$ | $0$ |
| 145.20.0.b.1 | $145$ | $2$ | $2$ | $0$ |
| 145.300.23.a.1 | $145$ | $30$ | $30$ | $23$ |
| 155.20.0.a.1 | $155$ | $2$ | $2$ | $0$ |
| 155.20.0.b.1 | $155$ | $2$ | $2$ | $0$ |
| 165.20.0.a.1 | $165$ | $2$ | $2$ | $0$ |
| 165.20.0.b.1 | $165$ | $2$ | $2$ | $0$ |
| 170.20.0.a.1 | $170$ | $2$ | $2$ | $0$ |
| 170.20.0.b.1 | $170$ | $2$ | $2$ | $0$ |
| 170.20.1.a.1 | $170$ | $2$ | $2$ | $1$ |
| 170.20.1.b.1 | $170$ | $2$ | $2$ | $1$ |
| 185.20.0.a.1 | $185$ | $2$ | $2$ | $0$ |
| 185.20.0.b.1 | $185$ | $2$ | $2$ | $0$ |
| 190.20.0.a.1 | $190$ | $2$ | $2$ | $0$ |
| 190.20.0.b.1 | $190$ | $2$ | $2$ | $0$ |
| 190.20.1.a.1 | $190$ | $2$ | $2$ | $1$ |
| 190.20.1.b.1 | $190$ | $2$ | $2$ | $1$ |
| 195.20.0.a.1 | $195$ | $2$ | $2$ | $0$ |
| 195.20.0.b.1 | $195$ | $2$ | $2$ | $0$ |
| 205.20.0.a.1 | $205$ | $2$ | $2$ | $0$ |
| 205.20.0.b.1 | $205$ | $2$ | $2$ | $0$ |
| 210.20.0.a.1 | $210$ | $2$ | $2$ | $0$ |
| 210.20.0.b.1 | $210$ | $2$ | $2$ | $0$ |
| 210.20.1.a.1 | $210$ | $2$ | $2$ | $1$ |
| 210.20.1.b.1 | $210$ | $2$ | $2$ | $1$ |
| 215.20.0.a.1 | $215$ | $2$ | $2$ | $0$ |
| 215.20.0.b.1 | $215$ | $2$ | $2$ | $0$ |
| 220.20.0.a.1 | $220$ | $2$ | $2$ | $0$ |
| 220.20.0.b.1 | $220$ | $2$ | $2$ | $0$ |
| 220.20.0.c.1 | $220$ | $2$ | $2$ | $0$ |
| 220.20.0.d.1 | $220$ | $2$ | $2$ | $0$ |
| 220.20.1.a.1 | $220$ | $2$ | $2$ | $1$ |
| 220.20.1.b.1 | $220$ | $2$ | $2$ | $1$ |
| 230.20.0.a.1 | $230$ | $2$ | $2$ | $0$ |
| 230.20.0.b.1 | $230$ | $2$ | $2$ | $0$ |
| 230.20.1.a.1 | $230$ | $2$ | $2$ | $1$ |
| 230.20.1.b.1 | $230$ | $2$ | $2$ | $1$ |
| 235.20.0.a.1 | $235$ | $2$ | $2$ | $0$ |
| 235.20.0.b.1 | $235$ | $2$ | $2$ | $0$ |
| 255.20.0.a.1 | $255$ | $2$ | $2$ | $0$ |
| 255.20.0.b.1 | $255$ | $2$ | $2$ | $0$ |
| 260.20.0.a.1 | $260$ | $2$ | $2$ | $0$ |
| 260.20.0.b.1 | $260$ | $2$ | $2$ | $0$ |
| 260.20.0.c.1 | $260$ | $2$ | $2$ | $0$ |
| 260.20.0.d.1 | $260$ | $2$ | $2$ | $0$ |
| 260.20.1.a.1 | $260$ | $2$ | $2$ | $1$ |
| 260.20.1.b.1 | $260$ | $2$ | $2$ | $1$ |
| 265.20.0.a.1 | $265$ | $2$ | $2$ | $0$ |
| 265.20.0.b.1 | $265$ | $2$ | $2$ | $0$ |
| 280.20.0.a.1 | $280$ | $2$ | $2$ | $0$ |
| 280.20.0.b.1 | $280$ | $2$ | $2$ | $0$ |
| 280.20.0.c.1 | $280$ | $2$ | $2$ | $0$ |
| 280.20.0.d.1 | $280$ | $2$ | $2$ | $0$ |
| 280.20.0.e.1 | $280$ | $2$ | $2$ | $0$ |
| 280.20.0.f.1 | $280$ | $2$ | $2$ | $0$ |
| 280.20.0.g.1 | $280$ | $2$ | $2$ | $0$ |
| 280.20.0.h.1 | $280$ | $2$ | $2$ | $0$ |
| 280.20.1.a.1 | $280$ | $2$ | $2$ | $1$ |
| 280.20.1.b.1 | $280$ | $2$ | $2$ | $1$ |
| 280.20.1.c.1 | $280$ | $2$ | $2$ | $1$ |
| 280.20.1.d.1 | $280$ | $2$ | $2$ | $1$ |
| 285.20.0.a.1 | $285$ | $2$ | $2$ | $0$ |
| 285.20.0.b.1 | $285$ | $2$ | $2$ | $0$ |
| 290.20.0.a.1 | $290$ | $2$ | $2$ | $0$ |
| 290.20.0.b.1 | $290$ | $2$ | $2$ | $0$ |
| 290.20.1.a.1 | $290$ | $2$ | $2$ | $1$ |
| 290.20.1.b.1 | $290$ | $2$ | $2$ | $1$ |
| 295.20.0.a.1 | $295$ | $2$ | $2$ | $0$ |
| 295.20.0.b.1 | $295$ | $2$ | $2$ | $0$ |
| 305.20.0.a.1 | $305$ | $2$ | $2$ | $0$ |
| 305.20.0.b.1 | $305$ | $2$ | $2$ | $0$ |
| 310.20.0.a.1 | $310$ | $2$ | $2$ | $0$ |
| 310.20.0.b.1 | $310$ | $2$ | $2$ | $0$ |
| 310.20.1.a.1 | $310$ | $2$ | $2$ | $1$ |
| 310.20.1.b.1 | $310$ | $2$ | $2$ | $1$ |
| 330.20.0.a.1 | $330$ | $2$ | $2$ | $0$ |
| 330.20.0.b.1 | $330$ | $2$ | $2$ | $0$ |
| 330.20.1.a.1 | $330$ | $2$ | $2$ | $1$ |
| 330.20.1.b.1 | $330$ | $2$ | $2$ | $1$ |
| 335.20.0.a.1 | $335$ | $2$ | $2$ | $0$ |
| 335.20.0.b.1 | $335$ | $2$ | $2$ | $0$ |