Invariants
| Level: | $20$ | $\SL_2$-level: | $10$ | Newform level: | $400$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot10^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.24.1.10 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}15&7\\11&16\end{bmatrix}$, $\begin{bmatrix}15&14\\13&9\end{bmatrix}$ |
| 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: | $1920$ |
Jacobian
| Conductor: | $2^{4}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 400.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + y z $ |
| $=$ | $11 x^{2} + 125 y^{2} - 11 y z + z^{2} + 5 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 125 x^{4} + 22 x^{2} z^{2} + 5 y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{118162368yz^{5}-7828380000yz^{3}w^{2}+267187500yzw^{4}-8895744z^{6}+1073051280z^{4}w^{2}-515475000z^{2}w^{4}+1953125w^{6}}{z(68381yz^{4}-751250yz^{2}w^{2}+390625yw^{4}-5148z^{5}+1760z^{3}w^{2}+137500zw^{4})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.12.0.a.1 | $10$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 20.12.0.m.2 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 20.12.1.b.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.72.1.j.2 | $20$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 20.96.5.h.1 | $20$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 20.120.5.w.1 | $20$ | $5$ | $5$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.72.5.cm.2 | $60$ | $3$ | $3$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.96.5.bq.1 | $60$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 100.120.5.f.1 | $100$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 140.192.13.i.1 | $140$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 220.288.21.i.2 | $220$ | $12$ | $12$ | $21$ | $?$ | not computed |