Invariants
| Level: | $20$ | $\SL_2$-level: | $10$ | Newform level: | $400$ | ||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $1 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot10$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.12.1.2 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}0&11\\17&16\end{bmatrix}$, $\begin{bmatrix}1&16\\6&5\end{bmatrix}$, $\begin{bmatrix}7&18\\13&19\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 20-isogeny field degree: | $6$ |
| Cyclic 20-torsion field degree: | $48$ |
| Full 20-torsion field degree: | $3840$ |
Jacobian
| Conductor: | $2^{4}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 400.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 908x + 15688 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(-37:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 12 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{5^5}\cdot\frac{70x^{2}y^{2}+625x^{2}z^{2}+2405xy^{2}z-5422500xz^{3}+y^{4}-5845y^{2}z^{2}-199535000z^{4}}{z^{3}(x+37z)}$ |
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 | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 20.2.0.a.1 | $20$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.24.1.e.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 20.24.1.e.2 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 20.24.1.f.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 20.24.1.f.2 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 20.36.1.d.1 | $20$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 20.48.3.h.1 | $20$ | $4$ | $4$ | $3$ | $0$ | $1^{2}$ |
| 20.60.3.k.1 | $20$ | $5$ | $5$ | $3$ | $0$ | $1^{2}$ |
| 40.24.1.co.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.co.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.cr.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.24.1.cr.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.24.1.ba.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.24.1.ba.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.24.1.bb.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.24.1.bb.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.3.c.1 | $60$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 60.48.3.ba.1 | $60$ | $4$ | $4$ | $3$ | $0$ | $1^{2}$ |
| 100.60.3.b.1 | $100$ | $5$ | $5$ | $3$ | $?$ | not computed |
| 120.24.1.je.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.je.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.jh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.jh.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 140.24.1.e.1 | $140$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 140.24.1.e.2 | $140$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 140.24.1.f.1 | $140$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 140.24.1.f.2 | $140$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 140.96.7.b.1 | $140$ | $8$ | $8$ | $7$ | $?$ | not computed |
| 140.252.19.b.1 | $140$ | $21$ | $21$ | $19$ | $?$ | not computed |
| 220.24.1.e.1 | $220$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 220.24.1.e.2 | $220$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 220.24.1.f.1 | $220$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 220.24.1.f.2 | $220$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 220.144.11.b.1 | $220$ | $12$ | $12$ | $11$ | $?$ | not computed |
| 260.24.1.e.1 | $260$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 260.24.1.e.2 | $260$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 260.24.1.f.1 | $260$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 260.24.1.f.2 | $260$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 260.168.13.b.1 | $260$ | $14$ | $14$ | $13$ | $?$ | not computed |
| 280.24.1.co.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.co.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.cr.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.24.1.cr.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |