Invariants
| Level: | $44$ | $\SL_2$-level: | $4$ | ||||
| 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 $4$ are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| 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: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 44.48.0.11 |
Level structure
| $\GL_2(\Z/44\Z)$-generators: | $\begin{bmatrix}3&8\\14&29\end{bmatrix}$, $\begin{bmatrix}31&4\\32&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.24.0.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: | $26400$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 61 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{4}-4x^{3}y+8x^{2}y^{2}+16xy^{3}+16y^{4})^{3}(x^{4}+4x^{3}y+8x^{2}y^{2}-16xy^{3}+16y^{4})^{3}}{y^{4}x^{28}(x-2y)^{4}(x+2y)^{4}(x^{2}+4y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 44.24.0-4.a.1.1 | $44$ | $2$ | $2$ | $0$ | $0$ |
| 44.24.0-4.a.1.2 | $44$ | $2$ | $2$ | $0$ | $0$ |
| 44.24.0-4.b.1.1 | $44$ | $2$ | $2$ | $0$ | $0$ |
| 44.24.0-4.b.1.2 | $44$ | $2$ | $2$ | $0$ | $0$ |
| 44.24.0-4.b.1.3 | $44$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 44.576.19-44.b.1.2 | $44$ | $12$ | $12$ | $19$ |
| 44.2640.96-44.c.1.2 | $44$ | $55$ | $55$ | $96$ |
| 44.2640.96-44.d.1.1 | $44$ | $55$ | $55$ | $96$ |
| 44.3168.115-44.b.1.1 | $44$ | $66$ | $66$ | $115$ |
| 88.96.0-8.a.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-8.a.1.10 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.a.1.2 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.a.1.12 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-8.b.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-8.b.1.7 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-8.b.2.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-8.b.2.6 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.b.1.2 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.b.1.12 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.b.2.6 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.b.2.11 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-8.c.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-8.c.1.10 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.c.1.6 | $88$ | $2$ | $2$ | $0$ |
| 88.96.0-88.c.1.11 | $88$ | $2$ | $2$ | $0$ |
| 88.96.1-8.g.1.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-8.g.1.11 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-8.g.2.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-8.h.1.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-8.h.1.11 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-8.h.2.1 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.n.1.3 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.n.2.3 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.n.2.16 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.o.1.3 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.o.2.3 | $88$ | $2$ | $2$ | $1$ |
| 88.96.1-88.o.2.16 | $88$ | $2$ | $2$ | $1$ |
| 88.96.2-8.a.1.1 | $88$ | $2$ | $2$ | $2$ |
| 88.96.2-8.a.1.11 | $88$ | $2$ | $2$ | $2$ |
| 88.96.2-88.a.1.5 | $88$ | $2$ | $2$ | $2$ |
| 88.96.2-88.a.1.24 | $88$ | $2$ | $2$ | $2$ |
| 132.144.4-12.b.1.1 | $132$ | $3$ | $3$ | $4$ |
| 132.192.3-12.b.1.1 | $132$ | $4$ | $4$ | $3$ |
| 220.240.8-20.b.1.1 | $220$ | $5$ | $5$ | $8$ |
| 220.288.7-20.b.1.1 | $220$ | $6$ | $6$ | $7$ |
| 220.480.15-20.b.1.2 | $220$ | $10$ | $10$ | $15$ |
| 264.96.0-24.a.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.a.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.a.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.a.1.23 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.b.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.b.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.b.2.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.b.2.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.b.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.b.1.23 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.b.2.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.b.2.18 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.c.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-24.c.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.c.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.c.1.18 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-24.n.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.n.2.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.n.2.7 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.n.1.9 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.n.2.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.n.2.22 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.o.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.o.2.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-24.o.2.7 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.o.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.o.2.4 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.o.2.22 | $264$ | $2$ | $2$ | $1$ |
| 264.96.2-24.a.1.2 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-24.a.1.18 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.a.1.6 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.a.1.41 | $264$ | $2$ | $2$ | $2$ |
| 308.384.11-28.b.1.1 | $308$ | $8$ | $8$ | $11$ |