Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | Newform level: | $1200$ | ||
| 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: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}23&34\\104&39\end{bmatrix}$, $\begin{bmatrix}73&66\\18&103\end{bmatrix}$, $\begin{bmatrix}91&48\\0&13\end{bmatrix}$, $\begin{bmatrix}95&96\\42&71\end{bmatrix}$, $\begin{bmatrix}103&26\\4&111\end{bmatrix}$, $\begin{bmatrix}117&80\\98&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.48.1.a.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $24$ |
| Cyclic 120-torsion field degree: | $768$ |
| Full 120-torsion field degree: | $368640$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1200.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} - 108x - 288 $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{5^6}\cdot\frac{80x^{2}y^{14}+1481250x^{2}y^{12}z^{2}+5349375000x^{2}y^{10}z^{4}+11823095703125x^{2}y^{8}z^{6}+15784326171875000x^{2}y^{6}z^{8}+14399671783447265625x^{2}y^{4}z^{10}+7759997940063476562500x^{2}y^{2}z^{12}+2439999420642852783203125x^{2}z^{14}+2980xy^{14}z+26887500xy^{12}z^{3}+85670859375xy^{10}z^{5}+168925000000000xy^{8}z^{7}+212067187500000000xy^{6}z^{9}+180001354064941406250xy^{4}z^{11}+94960015964508056640625xy^{2}z^{13}+26640002317428588867187500xz^{15}+y^{16}+71220y^{14}z^{2}+330768750y^{12}z^{4}+842411171875y^{10}z^{6}+1287399365234375y^{8}z^{8}+1358272509765625000y^{6}z^{10}+906431117401123046875y^{4}z^{12}+375040143680572509765625y^{2}z^{14}+57960055618286132812500000z^{16}}{z^{4}y^{4}(60x^{2}y^{6}+574375x^{2}y^{4}z^{2}+1013125000x^{2}y^{2}z^{4}+479970703125x^{2}z^{6}+1710xy^{6}z+9083750xy^{4}z^{3}+12747890625xy^{2}z^{5}+5280117187500xz^{7}+y^{8}+32090y^{6}z^{2}+82519375y^{4}z^{4}+61137265625y^{2}z^{6}+11522812500000z^{8})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 40.24.0-20.a.1.3 | $40$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.48.0-6.a.1.12 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0-20.a.1.3 | $40$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 120.48.0-6.a.1.7 | $120$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 120.192.1-60.e.1.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.e.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.e.2.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.e.2.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.e.3.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.e.3.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.e.4.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.e.4.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ln.1.24 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ln.1.25 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ln.2.24 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ln.2.25 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ln.3.17 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ln.3.32 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ln.4.17 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ln.4.32 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-60.a.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.c.1.44 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.c.1.54 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.g.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.g.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.i.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.i.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.o.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.o.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.o.2.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.o.2.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.s.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.s.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.s.2.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.s.2.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.dp.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.dp.1.29 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.dt.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.dt.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ed.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ed.1.29 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ej.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ej.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fl.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fl.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fl.2.21 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fl.2.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ge.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ge.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ge.2.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ge.2.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.5-60.a.1.14 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.480.17-60.e.1.16 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |