Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $100$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $5 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $10^{12}$ | Cusp orbits | $4^{3}$ | ||
| 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: | 10B5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.240.5.24 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}15&2\\2&41\end{bmatrix}$, $\begin{bmatrix}19&48\\58&23\end{bmatrix}$, $\begin{bmatrix}49&12\\12&55\end{bmatrix}$, $\begin{bmatrix}53&48\\48&17\end{bmatrix}$, $\begin{bmatrix}55&36\\56&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 10.120.5.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: | $9216$ |
Jacobian
| Conductor: | $2^{6}\cdot5^{10}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}$ |
| Newforms: | 50.2.a.a$^{2}$, 50.2.a.b$^{2}$, 100.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} - 3 x y - x z + x w + x t - 2 y^{2} + 2 y z + 2 z^{2} + w^{2} + 2 w t + t^{2} $ |
| $=$ | $2 x^{2} + 2 x y - 2 x z + x w + 2 x t - y z + y t + 4 z^{2} + w^{2} + 2 w t + 2 t^{2}$ | |
| $=$ | $x^{2} - 3 x y - 5 x z - x t - y w - y t + w^{2} - t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36980 x^{8} - 54180 x^{7} z - 2325 x^{6} y^{2} + 74885 x^{6} z^{2} + 2350 x^{5} y^{2} z - 42040 x^{5} z^{3} + \cdots + 5 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 10.60.3.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -2x-y+3z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4x+2y-z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x-2y+z$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}-3X^{3}Y-5X^{2}Y^{2}-4XY^{3}-2Y^{4}+3X^{3}Z-18X^{2}YZ-17XY^{2}Z+4Y^{3}Z-5X^{2}Z^{2}+17XYZ^{2}-6Y^{2}Z^{2}+4XZ^{3}+4YZ^{3}-2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 10.120.5.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 5z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 36980X^{8}-54180X^{7}Z-2325X^{6}Y^{2}+74885X^{6}Z^{2}+2350X^{5}Y^{2}Z-42040X^{5}Z^{3}+44X^{4}Y^{4}-2675X^{4}Y^{2}Z^{2}+22600X^{4}Z^{4}-52X^{3}Y^{4}Z+650X^{3}Y^{2}Z^{3}-1910X^{3}Z^{5}+56X^{2}Y^{4}Z^{2}-325X^{2}Y^{2}Z^{4}+660X^{2}Z^{6}-8XY^{4}Z^{3}-20XZ^{7}+4Y^{4}Z^{4}+5Z^{8} $ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(5)$ | $5$ | $12$ | $6$ | $0$ | $0$ | full Jacobian |
| 12.12.0-2.a.1.1 | $12$ | $20$ | $20$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.120.3-10.a.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.120.3-10.a.1.4 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.480.13-20.a.1.2 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{8}$ |
| 60.480.13-20.a.1.3 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{8}$ |
| 60.480.13-60.a.1.2 | $60$ | $2$ | $2$ | $13$ | $5$ | $1^{8}$ |
| 60.480.13-60.a.1.3 | $60$ | $2$ | $2$ | $13$ | $5$ | $1^{8}$ |
| 60.480.13-60.a.1.7 | $60$ | $2$ | $2$ | $13$ | $5$ | $1^{8}$ |
| 60.480.13-20.b.1.2 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
| 60.480.13-20.b.1.4 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
| 60.480.13-20.b.1.5 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
| 60.480.13-60.b.1.2 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 60.480.13-60.b.1.3 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 60.480.13-60.b.1.7 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 60.480.13-20.e.1.1 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 60.480.13-20.e.1.2 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 60.480.13-20.e.1.3 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 60.480.13-20.f.1.1 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
| 60.480.13-20.f.1.2 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
| 60.480.13-20.f.1.3 | $60$ | $2$ | $2$ | $13$ | $1$ | $1^{8}$ |
| 60.480.13-60.m.1.2 | $60$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
| 60.480.13-60.m.1.4 | $60$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
| 60.480.13-60.m.1.5 | $60$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
| 60.480.13-60.n.1.2 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 60.480.13-60.n.1.4 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 60.480.13-60.n.1.5 | $60$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 60.720.13-10.b.1.1 | $60$ | $3$ | $3$ | $13$ | $0$ | $1^{8}$ |
| 60.720.25-30.d.1.3 | $60$ | $3$ | $3$ | $25$ | $6$ | $1^{14}\cdot2^{3}$ |
| 60.960.29-30.a.1.7 | $60$ | $4$ | $4$ | $29$ | $1$ | $1^{24}$ |
| 120.480.13-40.a.1.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-40.a.1.6 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-40.a.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.a.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.a.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.a.1.11 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-40.d.1.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-40.d.1.6 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-40.d.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.d.1.9 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.d.1.11 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.d.1.13 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-40.m.1.3 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-40.m.1.4 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-40.m.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-40.p.1.3 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-40.p.1.4 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-40.p.1.5 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bk.1.8 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bk.1.9 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bk.1.14 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bn.1.6 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bn.1.7 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.480.13-120.bn.1.15 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |