Invariants
| Level: | $156$ | $\SL_2$-level: | $6$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1\cdot2\cdot3\cdot6$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6F0 |
Level structure
| $\GL_2(\Z/156\Z)$-generators: | $\begin{bmatrix}9&32\\22&155\end{bmatrix}$, $\begin{bmatrix}35&22\\12&43\end{bmatrix}$, $\begin{bmatrix}101&48\\2&91\end{bmatrix}$, $\begin{bmatrix}107&66\\0&53\end{bmatrix}$, $\begin{bmatrix}152&97\\17&120\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.12.0.a.1 for the level structure with $-I$) |
| Cyclic 156-isogeny field degree: | $28$ |
| Cyclic 156-torsion field degree: | $1344$ |
| Full 156-torsion field degree: | $5031936$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 9048 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{12}(x+2y)^{3}(x^{3}+6x^{2}y-84xy^{2}-568y^{3})^{3}}{y^{6}x^{12}(x-10y)(x+6y)^{3}(x+8y)^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 156.8.0-3.a.1.4 | $156$ | $3$ | $3$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 156.48.0-6.a.1.5 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-6.b.1.2 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-78.b.1.10 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-78.c.1.10 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-12.d.1.6 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-12.f.1.7 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-12.g.1.11 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-12.h.1.3 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-12.i.1.7 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-12.j.1.7 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.o.1.9 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.p.1.14 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.q.1.14 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.r.1.16 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.s.1.14 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.t.1.14 | $156$ | $2$ | $2$ | $0$ |
| 156.48.1-12.i.1.3 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1-12.j.1.3 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1-12.k.1.3 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1-12.l.1.5 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1-156.m.1.13 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1-156.n.1.13 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1-156.o.1.13 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1-156.p.1.13 | $156$ | $2$ | $2$ | $1$ |
| 156.72.0-6.a.1.4 | $156$ | $3$ | $3$ | $0$ |
| 156.336.11-78.a.1.31 | $156$ | $14$ | $14$ | $11$ |
| 312.48.0-24.p.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.y.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bw.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bx.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.ca.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.cb.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.cc.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.cd.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fm.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fn.1.11 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fo.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fp.1.21 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fq.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fr.1.23 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fs.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ft.1.19 | $312$ | $2$ | $2$ | $0$ |
| 312.48.1-24.eq.1.15 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-24.er.1.15 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-24.es.1.11 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-24.et.1.15 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.hk.1.14 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.hl.1.18 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.hm.1.12 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.hn.1.10 | $312$ | $2$ | $2$ | $1$ |