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: | $\begin{bmatrix}28&77\\15&104\end{bmatrix}$, $\begin{bmatrix}61&78\\108&7\end{bmatrix}$, $\begin{bmatrix}133&54\\58&137\end{bmatrix}$, $\begin{bmatrix}145&134\\106&123\end{bmatrix}$ |
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: | $672$ |
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 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.1.11 | $12$ | $2$ | $2$ | $0$ | $0$ |
156.24.0-6.a.1.2 | $156$ | $2$ | $2$ | $0$ | $?$ |
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.1.19 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.e.1.6 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.q.1.7 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.s.1.8 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.bk.1.5 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.bm.1.6 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.bp.1.8 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.bq.1.9 | $156$ | $2$ | $2$ | $1$ |
156.144.1-156.m.1.4 | $156$ | $3$ | $3$ | $1$ |
312.96.1-312.gf.1.8 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.jx.1.9 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bad.1.12 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.baj.1.13 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.byw.1.9 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bzc.1.8 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bzm.1.13 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bzp.1.8 | $312$ | $2$ | $2$ | $1$ |