Invariants
| Level: | $42$ | $\SL_2$-level: | $6$ | Newform level: | $1764$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $6^{4}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.48.1.4 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}5&33\\0&29\end{bmatrix}$, $\begin{bmatrix}25&15\\39&20\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 42.24.1.c.1 for the level structure with $-I$) |
| Cyclic 42-isogeny field degree: | $24$ |
| Cyclic 42-torsion field degree: | $288$ |
| Full 42-torsion field degree: | $12096$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}\cdot7^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1764.2.a.e |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 343 $ |
Rational points
This modular curve has infinitely many rational points, including 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3^3}{7^3}\cdot\frac{(y^{2}+343z^{2})(y^{2}+3087z^{2})^{3}}{z^{2}y^{6}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.24.0-3.a.1.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 21.24.0-3.a.1.1 | $21$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 42.16.0-42.b.1.1 | $42$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 42.16.0-42.b.1.2 | $42$ | $3$ | $3$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 42.144.1-42.e.1.2 | $42$ | $3$ | $3$ | $1$ | $1$ | dimension zero |
| 42.384.13-42.e.1.1 | $42$ | $8$ | $8$ | $13$ | $3$ | $1^{10}\cdot2$ |
| 42.1008.37-42.by.1.2 | $42$ | $21$ | $21$ | $37$ | $14$ | $1^{12}\cdot2^{10}\cdot4$ |
| 42.1344.49-42.bl.1.3 | $42$ | $28$ | $28$ | $49$ | $16$ | $1^{22}\cdot2^{11}\cdot4$ |
| 84.192.5-84.bu.1.3 | $84$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 126.144.3-126.p.1.2 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.144.3-126.p.1.4 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.144.3-126.s.1.2 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.144.3-126.s.1.4 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.144.3-126.y.1.3 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.144.3-126.y.1.4 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.144.3-126.bb.1.2 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.144.3-126.bd.1.3 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.144.3-126.bd.1.4 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.144.3-126.bg.1.2 | $126$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 126.144.4-126.p.1.2 | $126$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 126.144.4-126.p.1.4 | $126$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 126.144.5-126.c.1.2 | $126$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 210.240.9-210.x.1.1 | $210$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 210.288.9-210.by.1.7 | $210$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 210.480.17-210.dr.1.1 | $210$ | $10$ | $10$ | $17$ | $?$ | not computed |