Invariants
| Level: | $42$ | $\SL_2$-level: | $6$ | Newform level: | $36$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $6$ are rational) | Cusp widths | $6^{12}$ | Cusp orbits | $1^{6}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.144.1.6 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}17&38\\30&7\end{bmatrix}$, $\begin{bmatrix}35&18\\18&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.72.1.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: | $4032$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 36.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 1 $ |
Rational points
This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:1)$, $(2:3:1)$, $(-1:0:1)$, $(0:1:0)$, $(2:-3:1)$, $(0:-1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(y^{2}+3z^{2})^{3}(y^{6}+225y^{4}z^{2}-405y^{2}z^{4}+243z^{6})^{3}}{z^{2}y^{6}(y-3z)^{6}(y-z)^{2}(y+z)^{2}(y+3z)^{6}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X(2)$ | $2$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 21.24.0-3.a.1.1 | $21$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.48.0-6.a.1.1 | $42$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 42.48.0-6.a.1.2 | $42$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 42.48.1-6.a.1.1 | $42$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 42.72.0-6.a.1.1 | $42$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 42.72.0-6.a.1.2 | $42$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.1152.37-42.a.1.2 | $42$ | $8$ | $8$ | $37$ | $1$ | $1^{30}\cdot2^{3}$ |
| 42.3024.109-42.a.1.2 | $42$ | $21$ | $21$ | $109$ | $27$ | $1^{36}\cdot2^{34}\cdot4$ |
| 42.4032.145-42.a.1.2 | $42$ | $28$ | $28$ | $145$ | $28$ | $1^{66}\cdot2^{37}\cdot4$ |
| 84.288.3-12.a.1.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.288.3-12.a.1.11 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.288.3-84.a.1.2 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.288.3-84.a.1.14 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.288.5-12.a.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.a.1.5 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-12.b.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.b.1.4 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-12.e.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.e.1.3 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-12.f.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.f.1.4 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.7-12.o.1.2 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.288.7-12.o.1.8 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.288.7-84.o.1.7 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.288.7-84.o.1.16 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 126.432.7-18.a.1.1 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-18.a.1.2 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.a.1.3 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.a.1.4 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-18.b.1.1 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.b.1.3 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.b.1.4 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-18.c.1.1 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.c.1.3 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.c.1.4 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.10-18.a.1.1 | $126$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 126.432.10-18.a.1.2 | $126$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 126.432.13-18.a.1.1 | $126$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 168.288.3-24.a.1.2 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.3-24.a.1.19 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.3-168.a.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.3-168.a.1.25 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.5-24.a.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.a.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.d.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.d.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.m.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.m.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.p.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.p.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.7-168.co.1.12 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.co.1.29 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-24.cw.1.6 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-24.cw.1.15 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |