Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $3600$ | ||
| 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 | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-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: | 60.192.5.394 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}16&35\\53&24\end{bmatrix}$, $\begin{bmatrix}51&5\\56&57\end{bmatrix}$, $\begin{bmatrix}59&30\\18&49\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.96.5.ci.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $11520$ |
Jacobian
| Conductor: | $2^{18}\cdot3^{4}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 80.2.a.a, 80.2.c.a, 900.2.a.b, 3600.2.a.be |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} + 3 x y - x z + y^{2} + y z + w^{2} $ |
| $=$ | $x^{2} - x y - x t + y^{2} - y t + 3 z^{2} - w^{2} + t^{2}$ | |
| $=$ | $x^{2} + 3 x y + 2 x z + x w + y^{2} - 2 y z + y w - 3 z^{2} - 2 w^{2} - 2 w t$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 25 x^{4} y^{4} + 150 x^{4} y^{2} z^{2} + 225 x^{4} z^{4} - 50 x^{2} y^{6} - 150 x^{2} y^{4} z^{2} + \cdots + 23409 z^{8} $ |
Rational points
This modular curve has no real points, and therefore no 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 :
| $\displaystyle X$ | $=$ | $\displaystyle -3x-2y+3z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -4x-y-z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -2x+2y+2z$ |
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 60.96.5.ci.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 25X^{4}Y^{4}+150X^{4}Y^{2}Z^{2}+225X^{4}Z^{4}-50X^{2}Y^{6}-150X^{2}Y^{4}Z^{2}+810X^{2}Y^{2}Z^{4}+4590X^{2}Z^{6}+25Y^{8}+540Y^{6}Z^{2}+3294Y^{4}Z^{4}+6156Y^{2}Z^{6}+23409Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.96.3-20.i.2.6 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.48.1-60.bd.2.2 | $60$ | $4$ | $4$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 60.96.3-20.i.2.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.576.13-60.pq.2.6 | $60$ | $3$ | $3$ | $13$ | $3$ | $1^{4}\cdot2^{2}$ |
| 60.576.21-60.ho.2.1 | $60$ | $3$ | $3$ | $21$ | $6$ | $1^{8}\cdot2^{4}$ |
| 60.768.25-60.cm.1.10 | $60$ | $4$ | $4$ | $25$ | $5$ | $1^{10}\cdot2^{5}$ |
| 60.960.29-60.pi.1.1 | $60$ | $5$ | $5$ | $29$ | $6$ | $1^{12}\cdot2^{6}$ |