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: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12E0 |
Level structure
| $\GL_2(\Z/156\Z)$-generators: | $\begin{bmatrix}92&27\\87&62\end{bmatrix}$, $\begin{bmatrix}105&116\\46&17\end{bmatrix}$, $\begin{bmatrix}123&154\\44&73\end{bmatrix}$, $\begin{bmatrix}143&58\\102&121\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 156.24.0.p.1 for the level structure with $-I$) |
| Cyclic 156-isogeny field degree: | $28$ |
| Cyclic 156-torsion field degree: | $1344$ |
| 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 but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.9 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 156.24.0-6.a.1.7 | $156$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 156.96.1-156.a.1.16 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.e.1.2 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.q.1.3 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.s.1.7 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.bk.1.2 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.bm.1.7 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.bp.1.6 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.bq.1.10 | $156$ | $2$ | $2$ | $1$ |
| 156.144.1-156.m.1.2 | $156$ | $3$ | $3$ | $1$ |
| 312.96.1-312.gf.1.12 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.jx.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bad.1.14 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.baj.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.byw.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bzc.1.12 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bzm.1.7 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bzp.1.12 | $312$ | $2$ | $2$ | $1$ |