Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $36$ | ||
| Index: | $36$ | $\PSL_2$-index: | $18$ | ||||
| Genus: | $1 = 1 + \frac{ 18 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (all of which are rational) | Cusp widths | $3^{2}\cdot12$ | Cusp orbits | $1^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}7&78\\12&61\end{bmatrix}$, $\begin{bmatrix}10&119\\83&2\end{bmatrix}$, $\begin{bmatrix}25&52\\72&53\end{bmatrix}$, $\begin{bmatrix}48&97\\95&66\end{bmatrix}$, $\begin{bmatrix}58&113\\59&68\end{bmatrix}$, $\begin{bmatrix}76&11\\29&98\end{bmatrix}$, $\begin{bmatrix}81&94\\8&75\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.18.1.c.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| Cyclic 120-torsion field degree: | $1536$ |
| Full 120-torsion field degree: | $983040$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 36.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 1 $ |
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 18 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(4y^{2}-3z^{2})^{3}}{z^{4}(y-z)(y+z)}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $12$ | $6$ | $0$ | $0$ | full Jacobian |
| 40.12.0-4.c.1.3 | $40$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.12.0-4.c.1.3 | $40$ | $3$ | $3$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 120.72.1-12.i.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-60.i.1.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-12.j.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-60.j.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-12.m.1.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-60.m.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-12.n.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-60.n.1.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-24.be.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-120.be.1.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-24.bh.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-120.bh.1.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-24.bq.1.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-120.bq.1.30 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-24.bt.1.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1-120.bt.1.26 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.2-12.b.1.37 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.j.1.13 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-12.p.1.21 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-12.s.1.9 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.s.1.12 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-12.t.1.9 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.t.1.11 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-12.w.1.5 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.w.1.23 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-12.x.1.5 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.x.1.21 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-12.ba.1.7 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.ba.1.18 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-12.bb.1.5 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.bb.1.18 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.be.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-60.bf.1.20 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.bu.1.13 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cc.1.13 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cc.1.17 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cf.1.13 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cf.1.17 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.ci.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.ci.1.16 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cj.1.30 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cj.1.32 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.ck.1.10 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.ck.1.12 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cl.1.6 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cl.1.8 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cm.1.11 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cm.1.12 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cn.1.7 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cn.1.8 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.co.1.15 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.co.1.16 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.co.1.27 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cp.1.15 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cp.1.16 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cq.1.15 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cq.1.16 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cr.1.15 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cr.1.16 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cr.1.25 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cs.1.7 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cs.1.8 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.ct.1.11 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.ct.1.12 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cu.1.6 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cu.1.8 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cu.1.18 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cu.1.22 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cv.1.10 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cv.1.12 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cv.1.34 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cv.1.38 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cw.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cw.1.16 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cw.1.10 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cw.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cx.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.cx.1.16 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cx.1.10 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cx.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cy.1.10 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cy.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cz.1.10 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.cz.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.da.1.18 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.da.1.22 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.db.1.34 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.db.1.38 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dc.1.34 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dc.1.38 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dd.1.18 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dd.1.22 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.de.1.13 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.de.1.10 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.de.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.df.1.10 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.df.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dg.1.10 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dg.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.dh.1.13 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dh.1.10 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dh.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.di.1.34 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.di.1.38 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dj.1.18 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dj.1.22 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.dq.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dq.1.26 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-24.dt.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.dt.1.26 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.ec.1.30 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.72.2-120.ef.1.30 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.180.7-60.c.1.44 | $120$ | $5$ | $5$ | $7$ | $?$ | not computed |
| 120.216.7-60.c.1.70 | $120$ | $6$ | $6$ | $7$ | $?$ | not computed |
| 120.360.13-60.be.1.67 | $120$ | $10$ | $10$ | $13$ | $?$ | not computed |