Invariants
| Level: | $42$ | $\SL_2$-level: | $14$ | Newform level: | $14$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1\cdot2\cdot7\cdot14$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-7,-28$) |
Other labels
| Cummins and Pauli (CP) label: | 14C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.48.1.7 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}29&31\\26&17\end{bmatrix}$, $\begin{bmatrix}29&32\\6&19\end{bmatrix}$, $\begin{bmatrix}33&16\\16&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 14.24.1.a.1 for the level structure with $-I$) |
| Cyclic 42-isogeny field degree: | $4$ |
| Cyclic 42-torsion field degree: | $48$ |
| Full 42-torsion field degree: | $12096$ |
Jacobian
| Conductor: | $2\cdot7$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 14.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} + \left(x + 1\right) y $ | $=$ | $ x^{3} + 4x - 6 $ |
Rational points
This modular curve has 4 rational cusps and 2 rational CM points, but no other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | $(1:-1:1)$, $(2:-5:1)$, $(0:1:0)$, $(9:23:1)$ | ||
| 49.a2 | $-7$ | $-3375$ | $= -1 \cdot 3^{3} \cdot 5^{3}$ | $8.124$ | $(9:-33:1)$ |
| 49.a1 | $-28$ | $16581375$ | $= 3^{3} \cdot 5^{3} \cdot 17^{3}$ | $16.624$ | $(2:2:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{34x^{2}y^{7}-182x^{2}y^{6}z-11430x^{2}y^{5}z^{2}-154440x^{2}y^{4}z^{3}+672354x^{2}y^{3}z^{4}+10402236x^{2}y^{2}z^{5}-14609565x^{2}yz^{6}-121236750x^{2}z^{7}-xy^{8}-449xy^{7}z+810xy^{6}z^{2}+50814xy^{5}z^{3}+415863xy^{4}z^{4}+1872126xy^{3}z^{5}-30769146xy^{2}z^{6}-123710868xyz^{7}+342481419xz^{8}-y^{9}+6y^{8}z+3075y^{7}z^{2}+23652y^{6}z^{3}-233127y^{5}z^{4}-1986390y^{4}z^{5}-7202520y^{3}z^{6}+7585785y^{2}z^{7}+238368771yz^{8}+22654080z^{9}}{z^{2}(3x^{2}y^{5}+192x^{2}y^{4}z-1472x^{2}y^{3}z^{2}+3844x^{2}y^{2}z^{3}-19398x^{2}yz^{4}+36573x^{2}z^{5}+xy^{6}-74xy^{5}z-490xy^{4}z^{2}-1262xy^{3}z^{3}-13320xy^{2}z^{4}-18988xyz^{5}-122459xz^{6}-24y^{6}z+430y^{5}z^{2}+1547y^{4}z^{3}+10953y^{3}z^{4}+29698y^{2}z^{5}+57236yz^{6}+91866z^{7})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $16$ | $8$ | $0$ | $0$ | full Jacobian |
| 21.16.0-7.a.1.2 | $21$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 21.16.0-7.a.1.2 | $21$ | $3$ | $3$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 42.96.2-14.a.1.1 | $42$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 42.96.2-42.c.1.3 | $42$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 42.96.2-14.f.1.2 | $42$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 42.96.2-42.f.1.1 | $42$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 42.144.1-14.a.1.1 | $42$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 42.144.1-14.a.2.3 | $42$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 42.144.1-14.b.1.3 | $42$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 42.144.5-42.a.1.3 | $42$ | $3$ | $3$ | $5$ | $0$ | $2^{2}$ |
| 42.192.5-42.a.1.4 | $42$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 42.336.7-14.a.1.1 | $42$ | $7$ | $7$ | $7$ | $0$ | $1^{4}\cdot2$ |
| 84.96.2-28.a.1.2 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-28.b.1.7 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-28.c.1.11 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-28.g.1.1 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-28.h.1.7 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-28.i.1.7 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-84.i.1.10 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-84.j.1.8 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-84.k.1.8 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-84.q.1.10 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-84.r.1.8 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-84.s.1.8 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.3-28.a.1.7 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.96.3-28.b.1.7 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.96.3-28.c.1.7 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.96.3-28.d.1.11 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.96.3-84.ca.1.8 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.96.3-84.cb.1.8 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.96.3-84.ce.1.8 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.96.3-84.cf.1.8 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 126.144.1-126.k.1.2 | $126$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 126.144.1-126.k.2.2 | $126$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 126.144.1-126.l.1.2 | $126$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 126.144.1-126.l.2.8 | $126$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 126.144.1-126.m.1.8 | $126$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 126.144.1-126.m.2.6 | $126$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 168.96.2-56.b.1.4 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-56.c.1.4 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-56.d.1.7 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-56.e.1.7 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-56.l.1.4 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-56.m.1.4 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-56.n.1.7 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-56.o.1.7 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.p.1.19 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.q.1.19 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.r.1.15 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.s.1.15 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.bh.1.19 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.bi.1.19 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.bj.1.15 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.bk.1.15 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.3-56.a.1.7 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-56.b.1.7 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-56.c.1.7 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-56.d.1.7 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-168.gi.1.7 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-168.gj.1.7 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-168.go.1.7 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-168.gp.1.7 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.96.2-70.c.1.1 | $210$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 210.96.2-210.c.1.2 | $210$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 210.96.2-70.e.1.2 | $210$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 210.96.2-210.e.1.1 | $210$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 210.240.9-70.a.1.3 | $210$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 210.288.9-70.a.1.9 | $210$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 210.480.17-70.a.1.8 | $210$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 294.336.7-98.a.1.4 | $294$ | $7$ | $7$ | $7$ | $?$ | not computed |