Invariants
| Level: | $156$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (all of which are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12E0 |
Level structure
| $\GL_2(\Z/156\Z)$-generators: | $\begin{bmatrix}11&24\\42&5\end{bmatrix}$, $\begin{bmatrix}40&119\\105&134\end{bmatrix}$, $\begin{bmatrix}77&142\\74&33\end{bmatrix}$, $\begin{bmatrix}83&16\\72&115\end{bmatrix}$, $\begin{bmatrix}97&66\\36&115\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.g.1 for the level structure with $-I$) |
| Cyclic 156-isogeny field degree: | $14$ |
| Cyclic 156-torsion field degree: | $672$ |
| Full 156-torsion field degree: | $2515968$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 330 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{24}(3x^{2}-4y^{2})^{3}(3x^{6}-12x^{4}y^{2}+144x^{2}y^{4}-64y^{6})^{3}}{y^{4}x^{36}(x-2y)^{3}(x+2y)^{3}(3x-2y)(3x+2y)}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $12$ | $6$ | $0$ | $0$ |
| 52.12.0-4.c.1.1 | $52$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 52.12.0-4.c.1.1 | $52$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 156.96.0-12.c.1.3 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-12.c.1.6 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-12.c.2.3 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-12.c.2.6 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-12.c.3.4 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-12.c.3.5 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-12.c.4.4 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-12.c.4.5 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-156.c.1.3 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-156.c.1.14 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-156.c.2.7 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-156.c.2.10 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-156.c.3.5 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-156.c.3.12 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-156.c.4.6 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-156.c.4.11 | $156$ | $2$ | $2$ | $0$ |
| 156.96.1-12.b.1.11 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-12.h.1.2 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-12.k.1.4 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.k.1.7 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-12.l.1.2 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.l.1.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.o.1.3 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.p.1.8 | $156$ | $2$ | $2$ | $1$ |
| 156.144.1-12.f.1.1 | $156$ | $3$ | $3$ | $1$ |
| 312.96.0-24.bs.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bs.1.31 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bs.2.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bs.2.30 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bt.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bt.1.31 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bt.2.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bt.2.30 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.1.12 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.2.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.2.12 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.3.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.3.16 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.4.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.bu.4.16 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dq.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dq.1.60 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dq.2.17 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dq.2.48 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dr.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dr.1.60 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dr.2.17 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.dr.2.48 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.1.58 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.2.17 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.2.46 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.3.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.3.54 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.4.30 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ds.4.33 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-24.cg.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.es.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.ik.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.in.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iq.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iq.1.24 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.ir.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.ir.1.39 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.is.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.is.1.23 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.it.1.12 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.it.1.21 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iu.1.12 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iu.1.21 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iv.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iv.1.23 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iw.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.iw.1.23 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.ix.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.ix.1.24 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zc.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zf.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zo.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zr.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zu.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zu.1.64 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zv.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zv.1.56 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zw.1.29 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zw.1.36 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zx.1.21 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zx.1.44 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zy.1.21 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zy.1.44 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zz.1.29 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.zz.1.36 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.baa.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.baa.1.56 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bab.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bab.1.64 | $312$ | $2$ | $2$ | $1$ |
| 312.96.2-24.f.1.9 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.f.1.24 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.f.2.9 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.f.2.24 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.g.1.9 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.g.1.24 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.g.2.9 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.g.2.24 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.h.1.5 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.h.1.60 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.h.2.17 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.h.2.48 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.i.1.5 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.i.1.60 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.i.2.17 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.i.2.48 | $312$ | $2$ | $2$ | $2$ |