Invariants
Level: | $156$ | $\SL_2$-level: | $6$ | ||||
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 | $2^{3}\cdot6^{3}$ | 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: | 6I0 |
Level structure
$\GL_2(\Z/156\Z)$-generators: | $\begin{bmatrix}42&151\\61&36\end{bmatrix}$, $\begin{bmatrix}78&37\\89&130\end{bmatrix}$, $\begin{bmatrix}128&99\\99&26\end{bmatrix}$, $\begin{bmatrix}154&123\\123&82\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 12.24.0.h.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, including 85 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^3}\cdot\frac{(x+2y)^{24}(5x^{2}-12xy+36y^{2})^{3}(89x^{6}+108x^{5}y-3060x^{4}y^{2}-4320x^{3}y^{3}+54000x^{2}y^{4}-67392xy^{5}+25920y^{6})^{3}}{(x-2y)^{6}(x+2y)^{24}(x+6y)^{2}(x^{2}-12xy-12y^{2})^{6}(13x^{2}-60xy+36y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
78.24.0-6.a.1.2 | $78$ | $2$ | $2$ | $0$ | $?$ |
156.16.0-12.b.1.1 | $156$ | $3$ | $3$ | $0$ | $?$ |
156.24.0-6.a.1.1 | $156$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
156.96.1-12.j.1.3 | $156$ | $2$ | $2$ | $1$ |
156.96.1-12.l.1.5 | $156$ | $2$ | $2$ | $1$ |
156.96.1-12.n.1.3 | $156$ | $2$ | $2$ | $1$ |
156.96.1-12.p.1.5 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.s.1.2 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.t.1.3 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.ba.1.3 | $156$ | $2$ | $2$ | $1$ |
156.96.1-156.bb.1.2 | $156$ | $2$ | $2$ | $1$ |
156.144.1-12.l.1.2 | $156$ | $3$ | $3$ | $1$ |
312.96.1-24.ii.1.5 | $312$ | $2$ | $2$ | $1$ |
312.96.1-24.io.1.5 | $312$ | $2$ | $2$ | $1$ |
312.96.1-24.jb.1.5 | $312$ | $2$ | $2$ | $1$ |
312.96.1-24.jh.1.5 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bai.1.5 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.bal.1.5 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.blc.1.9 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.blf.1.9 | $312$ | $2$ | $2$ | $1$ |