Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $3600$ | ||
| Index: | $120$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $6 = 1 + \frac{ 120 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $20^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $8$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $5$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 6$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3$) |
Other labels
| Cummins and Pauli (CP) label: | 20B6 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.120.6.2 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}7&30\\45&17\end{bmatrix}$, $\begin{bmatrix}15&13\\59&40\end{bmatrix}$, $\begin{bmatrix}16&55\\35&4\end{bmatrix}$, $\begin{bmatrix}53&20\\5&19\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $768$ |
| Full 60-torsion field degree: | $18432$ |
Jacobian
| Conductor: | $2^{22}\cdot3^{6}\cdot5^{11}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{6}$ |
| Newforms: | 80.2.a.a, 400.2.a.a, 400.2.a.e, 900.2.a.b, 3600.2.a.bc, 3600.2.a.be |
Models
Canonical model in $\mathbb{P}^{ 5 }$ defined by 6 equations
| $ 0 $ | $=$ | $ x y + x u - y^{2} + y z + 2 y t + y u - z^{2} + 2 z w - z u $ |
| $=$ | $3 y^{2} + 3 y z - 3 z^{2} + z u - 2 w u + u^{2}$ | |
| $=$ | $x z - 2 x u - 2 y z + 2 y w + 2 y u + 2 z w + 2 z t - z u$ | |
| $=$ | $2 x y + 2 x z - 2 x u - 2 y^{2} + y z + y w - 2 y t - z^{2} + z w + z t - 2 w^{2} + w u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{8} + 20 x^{7} z + 75 x^{6} y^{2} + 10 x^{6} z^{2} + 120 x^{5} y^{2} z - 40 x^{5} z^{3} + \cdots + 5 z^{8} $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
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 20.60.3.w.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -5x-y+2z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -4y+3z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -2y-z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}Y^{2}-2XY^{3}+Y^{4}-2X^{3}Z+4X^{2}YZ-2XY^{2}Z-2Y^{3}Z+X^{2}Z^{2}+4XYZ^{2}+Y^{2}Z^{2}+2XZ^{3} $ |
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 |
|---|---|---|---|---|---|---|
| 20.60.3.w.1 | $20$ | $2$ | $2$ | $3$ | $2$ | $1^{3}$ |
| 30.30.1.b.1 | $30$ | $4$ | $4$ | $1$ | $1$ | $1^{5}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.240.15.cp.1 | $60$ | $2$ | $2$ | $15$ | $7$ | $1^{9}$ |
| 60.240.15.cs.1 | $60$ | $2$ | $2$ | $15$ | $6$ | $1^{9}$ |
| 60.240.15.dd.1 | $60$ | $2$ | $2$ | $15$ | $10$ | $1^{9}$ |
| 60.240.15.dg.1 | $60$ | $2$ | $2$ | $15$ | $8$ | $1^{9}$ |
| 60.240.15.gk.1 | $60$ | $2$ | $2$ | $15$ | $9$ | $1^{9}$ |
| 60.240.15.gn.1 | $60$ | $2$ | $2$ | $15$ | $7$ | $1^{9}$ |
| 60.240.15.go.1 | $60$ | $2$ | $2$ | $15$ | $7$ | $1^{9}$ |
| 60.240.15.gr.1 | $60$ | $2$ | $2$ | $15$ | $8$ | $1^{9}$ |
| 60.360.20.m.1 | $60$ | $3$ | $3$ | $20$ | $11$ | $1^{14}$ |
| 60.360.22.ej.1 | $60$ | $3$ | $3$ | $22$ | $13$ | $1^{14}\cdot2$ |
| 60.480.35.gz.1 | $60$ | $4$ | $4$ | $35$ | $17$ | $1^{29}$ |
| 120.240.15.yk.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.240.15.yt.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.240.15.bag.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.240.15.bap.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.240.15.bma.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.240.15.bmj.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.240.15.bmm.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 120.240.15.bmv.1 | $120$ | $2$ | $2$ | $15$ | $?$ | not computed |