Invariants
| Level: | $44$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 44.24.0.11 |
Level structure
| $\GL_2(\Z/44\Z)$-generators: | $\begin{bmatrix}27&42\\42&15\end{bmatrix}$, $\begin{bmatrix}31&42\\12&9\end{bmatrix}$, $\begin{bmatrix}35&28\\28&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.12.0.b.1 for the level structure with $-I$) |
| Cyclic 44-isogeny field degree: | $12$ |
| Cyclic 44-torsion field degree: | $240$ |
| Full 44-torsion field degree: | $52800$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 2569 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(256x^{4}-16x^{2}y^{2}+y^{4})^{3}}{y^{4}x^{16}(4x-y)^{2}(4x+y)^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 44.12.0-2.a.1.1 | $44$ | $2$ | $2$ | $0$ | $0$ |
| 44.12.0-4.c.1.1 | $44$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 44.48.0-4.b.1.1 | $44$ | $2$ | $2$ | $0$ |
| 88.48.0-8.c.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-8.d.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-8.d.2.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-8.e.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-8.e.2.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-8.h.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-8.i.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.48.1-8.c.1.9 | $88$ | $2$ | $2$ | $1$ |
| 88.48.1-8.d.1.8 | $88$ | $2$ | $2$ | $1$ |
| 132.48.0-12.c.1.3 | $132$ | $2$ | $2$ | $0$ |
| 132.72.2-12.b.1.5 | $132$ | $3$ | $3$ | $2$ |
| 132.96.1-12.b.1.10 | $132$ | $4$ | $4$ | $1$ |
| 220.48.0-20.c.1.2 | $220$ | $2$ | $2$ | $0$ |
| 220.120.4-20.b.1.2 | $220$ | $5$ | $5$ | $4$ |
| 220.144.3-20.b.1.4 | $220$ | $6$ | $6$ | $3$ |
| 220.240.7-20.b.1.3 | $220$ | $10$ | $10$ | $7$ |
| 264.48.0-24.e.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.h.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.h.2.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.i.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.i.2.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.l.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-24.m.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.1-24.c.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-24.d.1.4 | $264$ | $2$ | $2$ | $1$ |
| 308.48.0-28.c.1.3 | $308$ | $2$ | $2$ | $0$ |
| 308.192.5-28.b.1.7 | $308$ | $8$ | $8$ | $5$ |
| 308.504.16-28.b.1.3 | $308$ | $21$ | $21$ | $16$ |
| 44.48.0-44.c.1.2 | $44$ | $2$ | $2$ | $0$ |
| 44.288.9-44.b.1.1 | $44$ | $12$ | $12$ | $9$ |
| 44.1320.46-44.b.1.2 | $44$ | $55$ | $55$ | $46$ |
| 44.1320.46-44.f.1.4 | $44$ | $55$ | $55$ | $46$ |
| 44.1584.55-44.b.1.6 | $44$ | $66$ | $66$ | $55$ |
| 88.48.0-88.e.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.h.1.10 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.h.2.3 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.i.1.11 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.i.2.2 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.l.1.12 | $88$ | $2$ | $2$ | $0$ |
| 88.48.0-88.m.1.9 | $88$ | $2$ | $2$ | $0$ |
| 88.48.1-88.c.1.10 | $88$ | $2$ | $2$ | $1$ |
| 88.48.1-88.d.1.5 | $88$ | $2$ | $2$ | $1$ |
| 132.48.0-132.c.1.5 | $132$ | $2$ | $2$ | $0$ |
| 220.48.0-220.c.1.5 | $220$ | $2$ | $2$ | $0$ |
| 264.48.0-264.e.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.t.1.21 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.t.2.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.u.1.19 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.u.2.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.x.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.y.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.48.1-264.c.1.20 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.d.1.10 | $264$ | $2$ | $2$ | $1$ |
| 308.48.0-308.c.1.3 | $308$ | $2$ | $2$ | $0$ |