Invariants
Level: | $228$ | $\SL_2$-level: | $12$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (all of which are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{6}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $6$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12E0 |
Level structure
$\GL_2(\Z/228\Z)$-generators: | $\begin{bmatrix}74&187\\165&28\end{bmatrix}$, $\begin{bmatrix}90&89\\175&128\end{bmatrix}$, $\begin{bmatrix}113&22\\104&195\end{bmatrix}$, $\begin{bmatrix}180&107\\115&140\end{bmatrix}$, $\begin{bmatrix}209&72\\180&221\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 12.24.0.g.1 for the level structure with $-I$) |
Cyclic 228-isogeny field degree: | $20$ |
Cyclic 228-torsion field degree: | $1440$ |
Full 228-torsion field degree: | $11819520$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 330 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{24}(3x^{2}-4y^{2})^{3}(3x^{6}-12x^{4}y^{2}+144x^{2}y^{4}-64y^{6})^{3}}{y^{4}x^{36}(x-2y)^{3}(x+2y)^{3}(3x-2y)(3x+2y)}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
$X_0(3)$ | $3$ | $12$ | $6$ | $0$ | $0$ |
76.12.0-4.c.1.1 | $76$ | $4$ | $4$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
76.12.0-4.c.1.1 | $76$ | $4$ | $4$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
228.96.0-12.c.1.3 | $228$ | $2$ | $2$ | $0$ |
228.96.0-12.c.1.6 | $228$ | $2$ | $2$ | $0$ |
228.96.0-12.c.2.3 | $228$ | $2$ | $2$ | $0$ |
228.96.0-12.c.2.6 | $228$ | $2$ | $2$ | $0$ |
228.96.0-12.c.3.3 | $228$ | $2$ | $2$ | $0$ |
228.96.0-12.c.3.6 | $228$ | $2$ | $2$ | $0$ |
228.96.0-12.c.4.3 | $228$ | $2$ | $2$ | $0$ |
228.96.0-12.c.4.6 | $228$ | $2$ | $2$ | $0$ |
228.96.0-228.c.1.7 | $228$ | $2$ | $2$ | $0$ |
228.96.0-228.c.1.10 | $228$ | $2$ | $2$ | $0$ |
228.96.0-228.c.2.7 | $228$ | $2$ | $2$ | $0$ |
228.96.0-228.c.2.10 | $228$ | $2$ | $2$ | $0$ |
228.96.0-228.c.3.4 | $228$ | $2$ | $2$ | $0$ |
228.96.0-228.c.3.13 | $228$ | $2$ | $2$ | $0$ |
228.96.0-228.c.4.6 | $228$ | $2$ | $2$ | $0$ |
228.96.0-228.c.4.11 | $228$ | $2$ | $2$ | $0$ |
228.96.1-12.b.1.11 | $228$ | $2$ | $2$ | $1$ |
228.96.1-12.h.1.2 | $228$ | $2$ | $2$ | $1$ |
228.96.1-12.k.1.4 | $228$ | $2$ | $2$ | $1$ |
228.96.1-228.k.1.7 | $228$ | $2$ | $2$ | $1$ |
228.96.1-12.l.1.2 | $228$ | $2$ | $2$ | $1$ |
228.96.1-228.l.1.1 | $228$ | $2$ | $2$ | $1$ |
228.96.1-228.o.1.5 | $228$ | $2$ | $2$ | $1$ |
228.96.1-228.p.1.3 | $228$ | $2$ | $2$ | $1$ |
228.144.1-12.f.1.1 | $228$ | $3$ | $3$ | $1$ |