Invariants
| Level: | $120$ | $\SL_2$-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: | 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: | $\begin{bmatrix}7&8\\96&11\end{bmatrix}$, $\begin{bmatrix}27&20\\86&69\end{bmatrix}$, $\begin{bmatrix}33&34\\14&73\end{bmatrix}$, $\begin{bmatrix}87&106\\10&69\end{bmatrix}$, $\begin{bmatrix}99&64\\68&7\end{bmatrix}$, $\begin{bmatrix}113&24\\68&97\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.48.1.b.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: | 600.2.a.h |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{5^2}\cdot\frac{80x^{2}y^{14}-158981250x^{2}y^{12}z^{2}+21311949375000x^{2}y^{10}z^{4}-428572662939453125x^{2}y^{8}z^{6}+2296710849951171875000x^{2}y^{6}z^{8}-4759056729823150634765625x^{2}y^{4}z^{10}+4161181865265274047851562500x^{2}y^{2}z^{12}-1291401332105858325958251953125x^{2}z^{14}-20980xy^{14}z+8500387500xy^{12}z^{3}-708863458359375xy^{10}z^{5}+8925716073437500000xy^{8}z^{7}-36875883698730468750000xy^{6}z^{9}+64546168575401916503906250xy^{4}z^{11}-50077694644194126129150390625xy^{2}z^{13}+14205419071576566696166992187500xz^{15}-y^{16}+1385220y^{14}z^{2}-511796268750y^{12}z^{4}+17953003248671875y^{10}z^{6}-133872986579052734375y^{8}z^{8}+357270679475927734375000y^{6}z^{10}-409087002737379608154296875y^{4}z^{12}+198875928197747135162353515625y^{2}z^{14}-30993668317837600708007812500000z^{16}}{zy^{4}(1925x^{2}y^{8}z+400000x^{2}y^{6}z^{3}-500000000x^{2}y^{4}z^{5}+100000000000x^{2}y^{2}z^{7}+100000000000000x^{2}z^{9}+xy^{10}-20300xy^{8}z^{2}+7600000xy^{6}z^{4}-6000000000xy^{4}z^{6}+1900000000000xy^{2}z^{8}+900000000000000xz^{10}-73y^{10}z-60800y^{8}z^{3}+13600000y^{6}z^{5}+2500000000y^{4}z^{7}+13400000000000y^{2}z^{9}-3600000000000000z^{11})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 40.24.0-20.b.1.1 | $40$ | $4$ | $4$ | $0$ | $0$ | 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.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.24.0-20.b.1.1 | $40$ | $4$ | $4$ | $0$ | $0$ | 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.f.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.f.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.f.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.f.2.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.f.3.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.f.3.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.f.4.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.f.4.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lq.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lq.1.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lq.2.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lq.2.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lq.3.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lq.3.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lq.4.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lq.4.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-60.b.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.b.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.c.1.34 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.c.1.54 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.h.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.h.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.j.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.j.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.p.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.p.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.p.2.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.p.2.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.t.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.t.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.t.2.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.t.2.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.dr.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.dr.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.du.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.du.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eg.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eg.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.em.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.em.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fo.1.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fo.1.21 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fo.2.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fo.2.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.gh.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.gh.1.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.gh.2.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.gh.2.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.5-60.b.1.6 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.480.17-60.f.1.7 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |