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: | $\begin{bmatrix}49&107\\72&107\end{bmatrix}$, $\begin{bmatrix}51&1\\64&27\end{bmatrix}$, $\begin{bmatrix}67&51\\0&13\end{bmatrix}$, $\begin{bmatrix}91&11\\60&89\end{bmatrix}$, $\begin{bmatrix}93&1\\64&9\end{bmatrix}$, $\begin{bmatrix}97&80\\28&111\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.48.1.l.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $12$ |
| Cyclic 120-torsion field degree: | $384$ |
| 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{5^3}\cdot\frac{80x^{2}y^{14}-4818750x^{2}y^{12}z^{2}+88036875000x^{2}y^{10}z^{4}-246983154296875x^{2}y^{8}z^{6}-4441892822265625000x^{2}y^{6}z^{8}+1094319911041259765625x^{2}y^{4}z^{10}-89439631271362304687500x^{2}y^{2}z^{12}+2439999420642852783203125x^{2}z^{14}+2980xy^{14}z-179662500xy^{12}z^{3}+3416795859375xy^{10}z^{5}-17822307031250000xy^{8}z^{7}-45172169531250000000xy^{6}z^{9}+11736315453186035156250xy^{4}z^{11}-974241467189788818359375xy^{2}z^{13}+26640002317428588867187500xz^{15}+y^{16}-18780y^{14}z^{2}-1267181250y^{12}z^{4}+42106286171875y^{10}z^{6}-357415946728515625y^{8}z^{8}-21720038134765625000y^{6}z^{10}+19382122755584716796875y^{4}z^{12}-1957795452022552490234375y^{2}z^{14}+57960055618286132812500000z^{16}}{z^{2}y^{2}(70x^{2}y^{10}+1121250x^{2}y^{8}z^{2}+4204375000x^{2}y^{6}z^{4}+6398496093750x^{2}y^{4}z^{6}+4319890136718750x^{2}y^{2}z^{8}+1079995880126953125x^{2}z^{10}+2295xy^{10}z+21127500xy^{8}z^{3}+62850078125xy^{6}z^{5}+83207578125000xy^{4}z^{7}+51120494384765625xy^{2}z^{9}+11880016479492187500xz^{11}+y^{12}+51755y^{10}z^{2}+268572500y^{8}z^{4}+521351484375y^{6}z^{6}+464132705078125y^{4}z^{8}+186490162353515625y^{2}z^{10}+25920395507812500000z^{12})}$ |
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.h.1.2 | $40$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.48.0-12.g.1.13 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 120.48.0-12.g.1.3 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 40.24.0-20.h.1.2 | $40$ | $4$ | $4$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 120.192.1-60.m.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.m.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.m.2.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.m.2.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.m.3.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.m.3.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.m.4.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.m.4.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.288.5-60.dx.1.4 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.480.17-60.x.1.2 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 120.192.1-120.ru.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ru.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ru.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ru.2.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ru.3.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ru.3.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ru.4.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ru.4.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-120.ne.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ne.1.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ne.2.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ne.2.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ng.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ng.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ng.2.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ng.2.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.oc.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.oc.1.27 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.od.1.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.od.1.39 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.oo.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.oo.1.21 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.op.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.op.1.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pa.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pa.1.21 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pb.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pb.1.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pe.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pe.1.27 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pf.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pf.1.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ra.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ra.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ra.2.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ra.2.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.rc.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.rc.1.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.rc.2.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.rc.2.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |