Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $80$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $3 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $4^{2}\cdot20^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20B3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.96.3.187 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}8&45\\43&1\end{bmatrix}$, $\begin{bmatrix}31&50\\46&21\end{bmatrix}$, $\begin{bmatrix}36&25\\1&44\end{bmatrix}$, $\begin{bmatrix}56&25\\21&56\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.48.3.i.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $192$ |
| Full 60-torsion field degree: | $23040$ |
Jacobian
| Conductor: | $2^{12}\cdot5^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 80.2.a.a, 80.2.c.a |
Models
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ 2 x^{4} + 10 x^{3} z - 3 x^{2} y^{2} + 2 x^{2} y z + 5 x^{2} z^{2} - 12 x y^{2} z + 14 x y z^{2} + \cdots - 6 y z^{3} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:1:0)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^{11}\cdot5\,\frac{(y+z)^{3}(36x^{3}y^{6}+680x^{3}y^{5}z+9756x^{3}y^{4}z^{2}-2752x^{3}y^{3}z^{3}-4436x^{3}y^{2}z^{4}+8984x^{3}yz^{5}-14444x^{3}z^{6}-212x^{2}y^{7}-1052x^{2}y^{6}z-3292x^{2}y^{5}z^{2}+61932x^{2}y^{4}z^{3}-12604x^{2}y^{3}z^{4}+3084x^{2}y^{2}z^{5}+9260x^{2}yz^{6}-5212x^{2}z^{7}-776xy^{7}z-10936xy^{6}z^{2}-45576xy^{5}z^{3}+95656xy^{4}z^{4}-2632xy^{3}z^{5}+25512xy^{2}z^{6}-16360xyz^{7}-9176xz^{8}+375y^{9}+2951y^{8}z+9836y^{7}z^{2}-2724y^{6}z^{3}-38406y^{5}z^{4}+31482y^{4}z^{5}+15388y^{3}z^{6}+39660y^{2}z^{7}+12551yz^{8}+375z^{9})}{176x^{3}y^{9}-970x^{3}y^{8}z+5504x^{3}y^{7}z^{2}-19288x^{3}y^{6}z^{3}+32912x^{3}y^{5}z^{4}-30228x^{3}y^{4}z^{5}+160x^{3}y^{3}z^{6}+61048x^{3}y^{2}z^{7}-161856x^{3}yz^{8}+106334x^{3}z^{9}-217x^{2}y^{10}+1918x^{2}y^{9}z-11833x^{2}y^{8}z^{2}+51288x^{2}y^{7}z^{3}-141118x^{2}y^{6}z^{4}+233452x^{2}y^{5}z^{5}-166454x^{2}y^{4}z^{6}-76696x^{2}y^{3}z^{7}+57031x^{2}y^{2}z^{8}-68746x^{2}yz^{9}+43807x^{2}z^{10}-516xy^{10}z+5274xy^{9}z^{2}-30494xy^{8}z^{3}+123824xy^{7}z^{4}-308744xy^{6}z^{5}+370796xy^{5}z^{6}-50772xy^{4}z^{7}+75072xy^{3}z^{8}-367812xy^{2}z^{9}+68682xyz^{10}+69986xz^{11}-434y^{11}z+1126y^{10}z^{2}-3906y^{9}z^{3}-424y^{8}z^{4}+79044y^{7}z^{5}-195596y^{6}z^{6}+62372y^{5}z^{7}+119528y^{4}z^{8}+322062y^{3}z^{9}-147482y^{2}z^{10}-69986yz^{11}}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(4)$ | $4$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 15.24.0-5.a.1.2 | $15$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 15.24.0-5.a.1.2 | $15$ | $4$ | $4$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.192.5-20.a.1.5 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.192.5-20.c.1.5 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.192.5-20.e.2.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.192.5-20.g.2.4 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.192.5-60.v.2.4 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.192.5-60.x.2.6 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.192.5-60.cg.2.8 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.192.5-60.ci.2.6 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.288.7-20.bd.2.7 | $60$ | $3$ | $3$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 60.288.11-60.ba.2.18 | $60$ | $3$ | $3$ | $11$ | $3$ | $1^{2}\cdot2^{3}$ |
| 60.384.13-60.bg.2.3 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{2}$ |
| 60.480.15-20.bj.1.1 | $60$ | $5$ | $5$ | $15$ | $2$ | $1^{6}\cdot2^{3}$ |
| 120.192.5-40.b.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-40.h.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-40.n.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-40.t.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ij.2.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ip.2.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ber.2.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.bex.2.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.13-40.q.1.13 | $120$ | $4$ | $4$ | $13$ | $?$ | not computed |
| 300.480.15-100.i.1.7 | $300$ | $5$ | $5$ | $15$ | $?$ | not computed |