Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $50$ | ||
| Index: | $60$ | $\PSL_2$-index: | $30$ | ||||
| Genus: | $2 = 1 + \frac{ 30 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (all of which are rational) | Cusp widths | $10^{3}$ | Cusp orbits | $1^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 10B2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.60.2.12 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}13&54\\48&41\end{bmatrix}$, $\begin{bmatrix}23&40\\40&37\end{bmatrix}$, $\begin{bmatrix}35&44\\54&25\end{bmatrix}$, $\begin{bmatrix}49&46\\56&51\end{bmatrix}$, $\begin{bmatrix}59&8\\14&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 10.30.2.a.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $48$ |
| Cyclic 60-torsion field degree: | $768$ |
| Full 60-torsion field degree: | $36864$ |
Jacobian
| Conductor: | $2^{2}\cdot5^{4}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{2}$ |
| Newforms: | 50.2.a.b$^{2}$ |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ - x^{2} z + x y z - x z^{2} + x z w + 2 y^{3} - y z^{2} + y z w $ |
| $=$ | $2 x^{3} + x^{2} z - x y w - 2 y^{3} - y^{2} z + y^{2} w$ | |
| $=$ | $2 x^{2} y + 2 x y^{2} + x y z + 2 y^{3} + y^{2} z - y^{2} w$ | |
| $=$ | $2 x^{2} y - x^{2} z - 2 x y^{2} + y^{2} z - y^{2} w + z^{2} w - z w^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 2 x^{3} y + 4 x^{3} z + x^{2} y z + 6 x^{2} z^{2} - 2 x y^{2} z - x y z^{2} + 4 x z^{3} + \cdots + z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{2} + x\right) y $ | $=$ | $ x^{6} + 3x^{5} + 7x^{4} + 9x^{3} + 7x^{2} + 3x + 1 $ |
Rational points
This modular curve has 3 rational cusps and 1 rational CM point, but no other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Plane model | Weierstrass model | Embedded model | |
|---|---|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | |||||
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(0:1/2:1)$, $(-1:1:1)$, $(-2:1:0)$ | $(1:-1:0)$, $(0:1:1)$, $(-1:-1:1)$ | $(0:1:-1:1)$, $(-1/2:1/2:1/2:1)$, $(-1:0:2:1)$ |
Maps to other modular curves
$j$-invariant map of degree 30 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{2}-zw+w^{2})^{3}}{w^{2}z^{2}(z-w)^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 10.30.2.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+2X^{3}Y+4X^{3}Z+X^{2}YZ-2XY^{2}Z+6X^{2}Z^{2}-XYZ^{2}+4XZ^{3}-2YZ^{3}+Z^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 10.30.2.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x-y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -x^{3}-x^{2}y+xyw+y^{3}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $12$ | $6$ | $0$ | $0$ | full Jacobian |
| 12.12.0-2.a.1.2 | $12$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.12.0-2.a.1.2 | $12$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 60.20.0-10.a.1.1 | $60$ | $3$ | $3$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.120.4-20.a.1.4 | $60$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 60.120.4-60.a.1.7 | $60$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 60.120.4-20.b.1.6 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 60.120.4-60.b.1.7 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 60.120.4-20.c.1.4 | $60$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 60.120.4-60.c.1.6 | $60$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 60.120.4-20.d.1.4 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 60.120.4-60.d.1.8 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 60.180.4-10.a.1.6 | $60$ | $3$ | $3$ | $4$ | $0$ | $1^{2}$ |
| 60.180.7-30.a.1.11 | $60$ | $3$ | $3$ | $7$ | $3$ | $1^{5}$ |
| 60.240.5-10.b.1.3 | $60$ | $4$ | $4$ | $5$ | $0$ | $1^{3}$ |
| 60.240.8-30.a.1.20 | $60$ | $4$ | $4$ | $8$ | $0$ | $1^{6}$ |
| 120.120.4-40.a.1.6 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.a.1.12 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-40.b.1.3 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.b.1.11 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-40.c.1.6 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.c.1.11 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-40.d.1.3 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.d.1.10 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |