Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $600$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.96.1.10 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}13&34\\0&1\end{bmatrix}$, $\begin{bmatrix}17&0\\54&37\end{bmatrix}$, $\begin{bmatrix}23&30\\24&31\end{bmatrix}$, $\begin{bmatrix}29&30\\36&37\end{bmatrix}$, $\begin{bmatrix}31&42\\42&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.48.1.b.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $12$ |
| Cyclic 60-torsion field degree: | $192$ |
| Full 60-torsion field degree: | $23040$ |
Jacobian
| Conductor: | $2^{3}\cdot3\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 600.2.a.h |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 108x + 288 $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(8:0:1)$, $(3:0:1)$, $(-12:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{5^2}\cdot\frac{80x^{2}y^{14}-158981250x^{2}y^{12}z^{2}+21311949375000x^{2}y^{10}z^{4}-428572662939453125x^{2}y^{8}z^{6}+2296710849951171875000x^{2}y^{6}z^{8}-4759056729823150634765625x^{2}y^{4}z^{10}+4161181865265274047851562500x^{2}y^{2}z^{12}-1291401332105858325958251953125x^{2}z^{14}-20980xy^{14}z+8500387500xy^{12}z^{3}-708863458359375xy^{10}z^{5}+8925716073437500000xy^{8}z^{7}-36875883698730468750000xy^{6}z^{9}+64546168575401916503906250xy^{4}z^{11}-50077694644194126129150390625xy^{2}z^{13}+14205419071576566696166992187500xz^{15}-y^{16}+1385220y^{14}z^{2}-511796268750y^{12}z^{4}+17953003248671875y^{10}z^{6}-133872986579052734375y^{8}z^{8}+357270679475927734375000y^{6}z^{10}-409087002737379608154296875y^{4}z^{12}+198875928197747135162353515625y^{2}z^{14}-30993668317837600708007812500000z^{16}}{zy^{4}(1925x^{2}y^{8}z+400000x^{2}y^{6}z^{3}-500000000x^{2}y^{4}z^{5}+100000000000x^{2}y^{2}z^{7}+100000000000000x^{2}z^{9}+xy^{10}-20300xy^{8}z^{2}+7600000xy^{6}z^{4}-6000000000xy^{4}z^{6}+1900000000000xy^{2}z^{8}+900000000000000xz^{10}-73y^{10}z-60800y^{8}z^{3}+13600000y^{6}z^{5}+2500000000y^{4}z^{7}+13400000000000y^{2}z^{9}-3600000000000000z^{11})}$ |
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 |
|---|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
| 20.12.0.b.1 | $20$ | $8$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.48.0-6.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-6.a.1.7 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-60.q.1.2 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.0-60.q.1.15 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.48.1-60.x.1.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.48.1-60.x.1.15 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.192.1-60.f.1.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.192.1-60.f.2.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.192.1-60.f.3.4 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.192.1-60.f.4.4 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.192.3-60.b.1.12 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.192.3-60.c.1.12 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.192.3-60.h.1.8 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.192.3-60.j.1.16 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.192.3-60.p.1.14 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.192.3-60.p.2.16 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.192.3-60.t.1.14 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.192.3-60.t.2.16 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.288.5-60.b.1.1 | $60$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 60.480.17-60.f.1.20 | $60$ | $5$ | $5$ | $17$ | $1$ | $1^{16}$ |
| 60.576.17-60.f.1.36 | $60$ | $6$ | $6$ | $17$ | $1$ | $1^{16}$ |
| 60.960.33-60.r.1.40 | $60$ | $10$ | $10$ | $33$ | $1$ | $1^{32}$ |
| 120.192.1-120.lq.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lq.2.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lq.3.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lq.4.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-120.dr.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.du.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eg.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.em.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fo.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fo.2.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.gh.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.gh.2.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.288.5-180.b.1.2 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 180.288.9-180.b.1.18 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 180.288.9-180.f.1.6 | $180$ | $3$ | $3$ | $9$ | $?$ | not computed |