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: | $\begin{bmatrix}20&121\\131&108\end{bmatrix}$, $\begin{bmatrix}123&8\\62&15\end{bmatrix}$, $\begin{bmatrix}153&148\\52&21\end{bmatrix}$, $\begin{bmatrix}154&53\\97&132\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 156.24.0.s.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.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 78.24.0-6.a.1.2 | $78$ | $2$ | $2$ | $0$ | $?$ |
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.c.1.4 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.f.1.3 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.z.1.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.ba.1.3 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.bh.1.2 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.bi.1.2 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.bt.1.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.bu.1.11 | $156$ | $2$ | $2$ | $1$ |
| 156.144.1-156.r.1.6 | $156$ | $3$ | $3$ | $1$ |
| 312.96.1-312.gh.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.ka.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bla.1.5 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bld.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.byo.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.byr.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.bzy.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.cab.1.2 | $312$ | $2$ | $2$ | $1$ |