Invariants
| Level: | $42$ | $\SL_2$-level: | $6$ | ||||
| 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 | $1\cdot2\cdot3\cdot6$ | 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: | yes $\quad(D =$ $-3,-12$) |
Other labels
| Cummins and Pauli (CP) label: | 6F0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.24.0.5 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}23&2\\15&11\end{bmatrix}$, $\begin{bmatrix}23&20\\21&37\end{bmatrix}$, $\begin{bmatrix}29&0\\0&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.12.0.a.1 for the level structure with $-I$) |
| Cyclic 42-isogeny field degree: | $8$ |
| Cyclic 42-torsion field degree: | $96$ |
| Full 42-torsion field degree: | $24192$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 9048 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{12}(x+2y)^{3}(x^{3}+6x^{2}y-84xy^{2}-568y^{3})^{3}}{y^{6}x^{12}(x-10y)(x+6y)^{3}(x+8y)^{2}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $8$ | $4$ | $0$ | $0$ |
| 21.8.0-3.a.1.2 | $21$ | $3$ | $3$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 21.8.0-3.a.1.2 | $21$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 42.48.0-6.a.1.2 | $42$ | $2$ | $2$ | $0$ |
| 42.48.0-6.b.1.2 | $42$ | $2$ | $2$ | $0$ |
| 42.72.0-6.a.1.1 | $42$ | $3$ | $3$ | $0$ |
| 84.48.0-12.d.1.3 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-12.f.1.6 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-12.g.1.10 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-12.h.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-12.i.1.6 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-12.j.1.6 | $84$ | $2$ | $2$ | $0$ |
| 84.48.1-12.i.1.2 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1-12.j.1.2 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1-12.k.1.2 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1-12.l.1.4 | $84$ | $2$ | $2$ | $1$ |
| 126.72.0-18.a.1.2 | $126$ | $3$ | $3$ | $0$ |
| 126.72.2-18.c.1.3 | $126$ | $3$ | $3$ | $2$ |
| 126.72.2-18.d.1.3 | $126$ | $3$ | $3$ | $2$ |
| 168.48.0-24.p.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.y.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bw.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bx.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.ca.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.cb.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.cc.1.15 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.cd.1.15 | $168$ | $2$ | $2$ | $0$ |
| 168.48.1-24.eq.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-24.er.1.14 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-24.es.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-24.et.1.14 | $168$ | $2$ | $2$ | $1$ |
| 210.48.0-30.a.1.2 | $210$ | $2$ | $2$ | $0$ |
| 210.48.0-30.b.1.1 | $210$ | $2$ | $2$ | $0$ |
| 210.120.4-30.b.1.4 | $210$ | $5$ | $5$ | $4$ |
| 210.144.3-30.a.1.14 | $210$ | $6$ | $6$ | $3$ |
| 210.240.7-30.h.1.4 | $210$ | $10$ | $10$ | $7$ |
| 42.48.0-42.b.1.2 | $42$ | $2$ | $2$ | $0$ |
| 42.48.0-42.c.1.2 | $42$ | $2$ | $2$ | $0$ |
| 42.192.5-42.a.1.12 | $42$ | $8$ | $8$ | $5$ |
| 42.504.16-42.a.1.15 | $42$ | $21$ | $21$ | $16$ |
| 42.672.21-42.a.1.11 | $42$ | $28$ | $28$ | $21$ |
| 84.48.0-84.m.1.11 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.n.1.4 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.o.1.8 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.p.1.7 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.q.1.8 | $84$ | $2$ | $2$ | $0$ |
| 84.48.0-84.r.1.4 | $84$ | $2$ | $2$ | $0$ |
| 84.48.1-84.m.1.7 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1-84.n.1.3 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1-84.o.1.3 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1-84.p.1.7 | $84$ | $2$ | $2$ | $1$ |
| 168.48.0-168.fg.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fh.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fi.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fj.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fk.1.13 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fl.1.13 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fm.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fn.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.1-168.hk.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.hl.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.hm.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.hn.1.12 | $168$ | $2$ | $2$ | $1$ |
| 210.48.0-210.a.1.3 | $210$ | $2$ | $2$ | $0$ |
| 210.48.0-210.b.1.3 | $210$ | $2$ | $2$ | $0$ |