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: | $\begin{bmatrix}63&62\\94&137\end{bmatrix}$, $\begin{bmatrix}72&133\\29&58\end{bmatrix}$, $\begin{bmatrix}118&89\\145&138\end{bmatrix}$, $\begin{bmatrix}122&49\\147&40\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 156.24.0.q.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 |
---|---|---|---|---|---|
6.24.0-6.a.1.3 | $6$ | $2$ | $2$ | $0$ | $0$ |
156.24.0-6.a.1.9 | $156$ | $2$ | $2$ | $0$ | $?$ |
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.b.1.8 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.h.1.2 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.r.1.2 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.t.1.1 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.bl.1.1 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.bn.1.1 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.bo.1.2 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.br.1.2 | $156$ | $2$ | $2$ | $1$ |
156.144.1-156.n.1.8 | $156$ | $3$ | $3$ | $1$ |
312.96.1-312.gg.1.1 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.kg.1.1 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bag.1.1 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bam.1.1 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.byz.1.1 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bzf.1.1 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bzj.1.1 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bzs.1.1 | $312$ | $2$ | $2$ | $1$ |