Invariants
| Level: | $68$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 68.12.0.15 |
Level structure
| $\GL_2(\Z/68\Z)$-generators: | $\begin{bmatrix}1&0\\27&7\end{bmatrix}$, $\begin{bmatrix}45&16\\9&23\end{bmatrix}$, $\begin{bmatrix}51&26\\60&21\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 68-isogeny field degree: | $36$ |
| Cyclic 68-torsion field degree: | $1152$ |
| Full 68-torsion field degree: | $626688$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 19 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(x-6y)^{12}(11x^{2}-68xy+1292y^{2})^{3}(15x^{2}+204xy+1564y^{2})^{3}}{(x-6y)^{12}(x^{2}+68xy+68y^{2})^{4}(13x^{2}+68xy+1428y^{2})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.6.0.e.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 34.6.0.a.1 | $34$ | $2$ | $2$ | $0$ | $0$ |
| 68.6.0.b.1 | $68$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 68.24.0.g.1 | $68$ | $2$ | $2$ | $0$ |
| 68.24.0.h.1 | $68$ | $2$ | $2$ | $0$ |
| 68.216.15.be.1 | $68$ | $18$ | $18$ | $15$ |
| 68.1632.121.bu.1 | $68$ | $136$ | $136$ | $121$ |
| 68.1836.136.bu.1 | $68$ | $153$ | $153$ | $136$ |
| 136.24.0.cc.1 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0.cd.1 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0.ck.1 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0.cl.1 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0.cq.1 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0.cr.1 | $136$ | $2$ | $2$ | $0$ |
| 136.24.1.y.1 | $136$ | $2$ | $2$ | $1$ |
| 136.24.1.ba.1 | $136$ | $2$ | $2$ | $1$ |
| 136.24.1.dk.1 | $136$ | $2$ | $2$ | $1$ |
| 136.24.1.dm.1 | $136$ | $2$ | $2$ | $1$ |
| 204.24.0.u.1 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0.v.1 | $204$ | $2$ | $2$ | $0$ |
| 204.36.2.fk.1 | $204$ | $3$ | $3$ | $2$ |
| 204.48.1.bq.1 | $204$ | $4$ | $4$ | $1$ |