Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | Newform level: | $24$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (all of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $1^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}5&36\\8&115\end{bmatrix}$, $\begin{bmatrix}17&88\\80&111\end{bmatrix}$, $\begin{bmatrix}47&112\\116&33\end{bmatrix}$, $\begin{bmatrix}51&118\\4&9\end{bmatrix}$, $\begin{bmatrix}63&88\\16&9\end{bmatrix}$, $\begin{bmatrix}67&110\\0&59\end{bmatrix}$, $\begin{bmatrix}93&100\\104&91\end{bmatrix}$, $\begin{bmatrix}113&78\\84&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.48.1.b.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: | 24.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} - 4x + 4 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{16x^{2}y^{14}-254370x^{2}y^{12}z^{2}+272792952x^{2}y^{10}z^{4}-43885840685x^{2}y^{8}z^{6}+1881465528280x^{2}y^{6}z^{8}-31188954184569x^{2}y^{4}z^{10}+218165771777620x^{2}y^{2}z^{12}-541652977285693x^{2}z^{14}-852xy^{14}z+2923620xy^{12}z^{3}-2032924815xy^{10}z^{5}+217907337732xy^{8}z^{7}-7546917207824xy^{6}z^{9}+109553117422806xy^{4}z^{11}-699635304814401xy^{2}z^{13}+1624959302500352xz^{15}-y^{16}+11420y^{14}z^{2}-33883710y^{12}z^{4}+9961460717y^{10}z^{6}-628484953611y^{8}z^{8}+14424778023672y^{6}z^{10}-146070717545179y^{4}z^{12}+662020649013103y^{2}z^{14}-1083306712635148z^{16}}{zy^{4}(77x^{2}y^{8}z+128x^{2}y^{6}z^{3}-1280x^{2}y^{4}z^{5}+2048x^{2}y^{2}z^{7}+16384x^{2}z^{9}+xy^{10}-224xy^{8}z^{2}+384xy^{6}z^{4}-2048xy^{4}z^{6}+6144xy^{2}z^{8}+16384xz^{10}-15y^{10}z-20y^{8}z^{3}+1280y^{4}z^{7}+8192y^{2}z^{9}-32768z^{11})}$ |
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 |
| 40.24.0-4.b.1.11 | $40$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.24.0-4.b.1.11 | $40$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 120.48.0-6.a.1.7 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.48.0-12.g.1.3 | $120$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 120.192.1-12.b.1.21 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-12.b.2.22 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-12.b.3.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-12.b.4.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.b.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.b.2.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.b.3.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.b.4.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.ck.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.ck.2.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.ck.3.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-24.ck.4.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lg.1.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lg.2.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lg.3.18 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lg.4.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-12.b.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.c.1.54 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.e.1.31 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.f.1.21 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.f.1.54 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.g.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-12.g.2.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.l.1.31 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.l.2.31 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bf.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bk.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bk.2.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bl.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bl.2.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bm.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bm.2.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bm.3.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bm.4.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bn.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bn.2.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bo.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bo.2.40 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bp.1.36 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bp.2.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bq.1.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bq.2.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.br.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.br.2.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.br.3.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.br.4.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bv.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.bv.2.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.cg.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.ch.1.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.ck.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-24.cl.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ds.1.59 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.dz.1.31 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eq.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eq.2.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.er.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.er.2.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.es.1.102 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.es.2.78 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.es.3.78 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.es.4.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.et.1.102 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.et.2.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eu.1.74 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eu.2.65 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ev.1.102 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ev.2.97 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ew.1.116 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ew.2.65 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ex.1.114 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ex.2.106 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ex.3.106 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ex.4.90 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fb.1.36 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fb.2.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fy.1.62 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fz.1.71 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.gc.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.gd.1.62 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.5-24.c.1.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.c.1.67 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-24.d.1.35 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.d.1.71 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-24.g.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.g.1.73 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-24.h.1.16 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.h.1.76 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-24.k.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-24.k.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.k.1.64 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.k.2.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-24.l.1.18 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-24.l.2.18 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.l.1.72 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.l.2.64 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-12.b.1.32 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.480.17-60.b.1.25 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |