Invariants
| Level: | $20$ | $\SL_2$-level: | $10$ | Newform level: | $400$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot10^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 10D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.48.1.11 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}14&11\\17&5\end{bmatrix}$, $\begin{bmatrix}17&11\\10&19\end{bmatrix}$ |
| $\GL_2(\Z/20\Z)$-subgroup: | $D_5.\GL(2,\mathbb{Z}/4)$ |
| Contains $-I$: | no $\quad$ (see 20.24.1.e.1 for the level structure with $-I$) |
| Cyclic 20-isogeny field degree: | $6$ |
| Cyclic 20-torsion field degree: | $24$ |
| Full 20-torsion field degree: | $960$ |
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} - 1033x + 12438 $ |
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 |
|---|
| $(18:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{5^5}\cdot\frac{40x^{2}y^{6}-975000x^{2}y^{4}z^{2}+7364375000x^{2}y^{2}z^{4}-18316640625000x^{2}z^{6}-1840xy^{6}z+37950000xy^{4}z^{3}-274286250000xy^{2}z^{5}+667657656250000xz^{7}-y^{8}+37660y^{6}z^{2}-498793750y^{4}z^{4}+2881368437500y^{2}z^{6}-6083246494140625z^{8}}{z^{3}y^{2}(3050x^{2}z+xy^{2}-111175xz^{2}-73y^{2}z+1012950z^{3})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.24.0-5.a.1.1 | $10$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 20.24.0-5.a.1.1 | $20$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 20.144.1-20.i.1.1 | $20$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 20.192.5-20.g.2.3 | $20$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 20.240.5-20.u.1.2 | $20$ | $5$ | $5$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.144.5-60.cl.2.6 | $60$ | $3$ | $3$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.192.5-60.bp.2.5 | $60$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 100.240.5-100.e.1.4 | $100$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 140.384.13-140.h.2.7 | $140$ | $8$ | $8$ | $13$ | $?$ | not computed |