Invariants
| Level: | $20$ | $\SL_2$-level: | $20$ | Newform level: | $400$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot20^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20D5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.192.5.21 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}1&18\\19&15\end{bmatrix}$, $\begin{bmatrix}8&19\\3&12\end{bmatrix}$ |
| $\GL_2(\Z/20\Z)$-subgroup: | $D_{30}:C_4$ |
| Contains $-I$: | no $\quad$ (see 20.96.5.g.1 for the level structure with $-I$) |
| Cyclic 20-isogeny field degree: | $6$ |
| Cyclic 20-torsion field degree: | $24$ |
| Full 20-torsion field degree: | $240$ |
Jacobian
| Conductor: | $2^{18}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 80.2.a.a, 80.2.c.a, 100.2.a.a, 400.2.a.c |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} + y z - y t + z t + w^{2} - t^{2} $ |
| $=$ | $x^{2} + x y - x z + 2 x t + y^{2} - 2 y z - y t + z^{2} + z t - t^{2}$ | |
| $=$ | $2 x^{2} - y^{2} + 2 y z + y w + y t - z^{2} + z w - z t - w^{2} + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 289 x^{8} - 170 x^{6} y^{2} - 228 x^{6} z^{2} + 25 x^{4} y^{4} + 90 x^{4} y^{2} z^{2} + 366 x^{4} z^{4} + \cdots + 25 z^{8} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 20.48.3.i.2 :
| $\displaystyle X$ | $=$ | $\displaystyle 3y-2z-3w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4y-z+w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2y+2z-2w$ |
Equation of the image curve:
| $0$ | $=$ | $ 6X^{3}Y-22X^{2}Y^{2}+6XY^{3}+12X^{2}YZ+14XY^{2}Z-6Y^{3}Z-3X^{2}Z^{2}-2XYZ^{2}+5Y^{2}Z^{2}-10YZ^{3}+2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.96.5.g.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y-z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 289X^{8}-170X^{6}Y^{2}-228X^{6}Z^{2}+25X^{4}Y^{4}+90X^{4}Y^{2}Z^{2}+366X^{4}Z^{4}-50X^{2}Y^{4}Z^{2}+50X^{2}Y^{2}Z^{4}-180X^{2}Z^{6}+25Y^{4}Z^{4}-50Y^{2}Z^{6}+25Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.48.1-20.e.2.2 | $20$ | $4$ | $4$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 20.96.3-20.i.2.3 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 20.96.3-20.i.2.6 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.576.13-20.bm.2.3 | $20$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{2}$ |
| 20.960.29-20.bs.1.2 | $20$ | $5$ | $5$ | $29$ | $2$ | $1^{12}\cdot2^{6}$ |
| 40.768.25-40.y.2.7 | $40$ | $4$ | $4$ | $25$ | $3$ | $1^{8}\cdot2^{4}\cdot4$ |
| 60.576.21-60.gy.2.6 | $60$ | $3$ | $3$ | $21$ | $4$ | $1^{8}\cdot2^{4}$ |
| 60.768.25-60.ce.1.5 | $60$ | $4$ | $4$ | $25$ | $1$ | $1^{10}\cdot2^{5}$ |