Invariants
| Level: | $42$ | $\SL_2$-level: | $6$ | Newform level: | $1764$ | ||
| 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 $2$ are rational) | Cusp widths | $6^{12}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| 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: | 6F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.144.1.5 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}5&24\\36&25\end{bmatrix}$, $\begin{bmatrix}41&12\\27&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 42.72.1.e.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: | $2^{2}\cdot3^{2}\cdot7^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1764.2.a.e |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.72.0-6.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 42.48.0-42.c.1.1 | $42$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 42.48.0-42.c.1.2 | $42$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 42.48.1-42.c.1.2 | $42$ | $3$ | $3$ | $1$ | $1$ | dimension zero |
| 42.72.0-6.a.1.1 | $42$ | $2$ | $2$ | $0$ | $0$ | 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.bb.1.4 | $42$ | $8$ | $8$ | $37$ | $7$ | $1^{30}\cdot2^{3}$ |
| 42.3024.109-42.cr.1.2 | $42$ | $21$ | $21$ | $109$ | $32$ | $1^{36}\cdot2^{34}\cdot4$ |
| 42.4032.145-42.eg.1.1 | $42$ | $28$ | $28$ | $145$ | $38$ | $1^{66}\cdot2^{37}\cdot4$ |
| 84.288.5-84.cf.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.cj.1.8 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.ds.1.3 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.dw.1.4 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.fo.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.fs.1.4 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.gv.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.gz.1.4 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 126.432.7-126.ce.1.3 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.ce.1.4 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.ch.1.3 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.ch.1.4 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.co.1.3 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.co.1.4 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.cs.1.2 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.cu.1.3 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.cu.1.4 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.7-126.cy.1.2 | $126$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 126.432.10-126.cu.1.2 | $126$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 126.432.10-126.cu.1.4 | $126$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 126.432.13-126.bx.1.2 | $126$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 168.288.5-168.qt.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.rv.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bdr.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bet.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bqm.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bro.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bzk.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.cam.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |