Invariants
Level: | $68$ | $\SL_2$-level: | $68$ | Newform level: | $1156$ | ||
Index: | $612$ | $\PSL_2$-index: | $306$ | ||||
Genus: | $22 = 1 + \frac{ 306 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 9 }{2}$ | ||||||
Cusps: | $9$ (of which $1$ is rational) | Cusp widths | $34^{9}$ | Cusp orbits | $1\cdot8$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $6$ | ||||||
$\Q$-gonality: | $6 \le \gamma \le 14$ | ||||||
$\overline{\Q}$-gonality: | $6 \le \gamma \le 14$ | ||||||
Rational cusps: | $1$ | ||||||
Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
Cummins and Pauli (CP) label: | 34A22 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 68.612.22.5 |
Level structure
$\GL_2(\Z/68\Z)$-generators: | $\begin{bmatrix}9&41\\29&8\end{bmatrix}$, $\begin{bmatrix}19&64\\28&15\end{bmatrix}$, $\begin{bmatrix}23&42\\20&45\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 34.306.22.a.1 for the level structure with $-I$) |
Cyclic 68-isogeny field degree: | $12$ |
Cyclic 68-torsion field degree: | $384$ |
Full 68-torsion field degree: | $12288$ |
Jacobian
Conductor: | $2^{30}\cdot17^{41}$ |
Simple: | no |
Squarefree: | yes |
Decomposition: | $1^{2}\cdot2^{5}\cdot3^{2}\cdot4$ |
Newforms: | 17.2.a.a, 68.2.a.a, 289.2.a.a, 289.2.a.b, 289.2.a.d, 1156.2.a.a, 1156.2.a.c, 1156.2.a.d, 1156.2.a.f, 1156.2.a.h |
Rational points
This modular curve has 1 rational cusp and 1 rational CM point, but no other known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
68.4.0-2.a.1.1 | $68$ | $153$ | $153$ | $0$ | $0$ | 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 |
---|---|---|---|---|---|---|
68.1224.43-34.a.1.3 | $68$ | $2$ | $2$ | $43$ | $15$ | $1\cdot2^{3}\cdot3^{2}\cdot4^{2}$ |
68.1224.43-68.a.1.1 | $68$ | $2$ | $2$ | $43$ | $25$ | $1\cdot2^{3}\cdot3^{2}\cdot4^{2}$ |
68.1224.43-34.b.1.1 | $68$ | $2$ | $2$ | $43$ | $16$ | $1\cdot2^{3}\cdot3^{2}\cdot4^{2}$ |
68.1224.43-68.d.1.2 | $68$ | $2$ | $2$ | $43$ | $14$ | $1\cdot2^{3}\cdot3^{2}\cdot4^{2}$ |
68.1836.64-34.a.1.5 | $68$ | $3$ | $3$ | $64$ | $28$ | $1^{8}\cdot2^{8}\cdot3^{6}$ |
68.2448.94-68.a.1.3 | $68$ | $4$ | $4$ | $94$ | $40$ | $1^{10}\cdot2^{17}\cdot3^{6}\cdot4\cdot6$ |