Invariants
Level: | $68$ | $\SL_2$-level: | $68$ | Newform level: | $2312$ | ||
Index: | $3672$ | $\PSL_2$-index: | $1836$ | ||||
Genus: | $136 = 1 + \frac{ 1836 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 36 }{2}$ | ||||||
Cusps: | $36$ (of which $4$ are rational) | Cusp widths | $34^{18}\cdot68^{18}$ | Cusp orbits | $1^{4}\cdot8^{2}\cdot16$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $57$ | ||||||
$\Q$-gonality: | $32 \le \gamma \le 84$ | ||||||
$\overline{\Q}$-gonality: | $32 \le \gamma \le 84$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 68.3672.136.88 |
Level structure
$\GL_2(\Z/68\Z)$-generators: | $\begin{bmatrix}9&50\\56&61\end{bmatrix}$, $\begin{bmatrix}13&54\\36&27\end{bmatrix}$, $\begin{bmatrix}15&66\\26&19\end{bmatrix}$, $\begin{bmatrix}61&44\\6&41\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 68.1836.136.g.1 for the level structure with $-I$) |
Cyclic 68-isogeny field degree: | $2$ |
Cyclic 68-torsion field degree: | $64$ |
Full 68-torsion field degree: | $2048$ |
Jacobian
Conductor: | $2^{211}\cdot17^{257}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{14}\cdot2^{19}\cdot3^{14}\cdot4^{6}\cdot6\cdot12$ |
Newforms: | 17.2.a.a$^{4}$, 34.2.a.a$^{3}$, 68.2.a.a$^{2}$, 136.2.a.a, 136.2.a.b, 136.2.a.c, 289.2.a.a$^{3}$, 289.2.a.b$^{3}$, 289.2.a.c, 289.2.a.d$^{3}$, 289.2.a.e, 289.2.a.f, 578.2.a.a$^{2}$, 578.2.a.b$^{2}$, 578.2.a.c$^{2}$, 578.2.a.d$^{2}$, 578.2.a.e$^{2}$, 578.2.a.f, 578.2.a.g, 578.2.a.h$^{2}$, 578.2.a.i, 1156.2.a.a, 1156.2.a.b, 1156.2.a.c, 1156.2.a.d, 1156.2.a.e, 1156.2.a.f, 1156.2.a.g, 1156.2.a.h, 2312.2.a.e, 2312.2.a.l, 2312.2.a.o, 2312.2.a.p, 2312.2.a.s, 2312.2.a.t, 2312.2.a.v, 2312.2.a.w |
Rational points
This modular curve has 4 rational cusps 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 |
---|---|---|---|---|---|---|
68.1836.64-34.a.1.4 | $68$ | $2$ | $2$ | $64$ | $28$ | $1^{4}\cdot2^{6}\cdot3^{6}\cdot4^{5}\cdot6\cdot12$ |
68.1836.64-34.a.1.5 | $68$ | $2$ | $2$ | $64$ | $28$ | $1^{4}\cdot2^{6}\cdot3^{6}\cdot4^{5}\cdot6\cdot12$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
68.7344.271-68.b.1.7 | $68$ | $2$ | $2$ | $271$ | $98$ | $1^{15}\cdot2^{22}\cdot3^{14}\cdot4^{7}\cdot6$ |
68.7344.271-68.c.1.4 | $68$ | $2$ | $2$ | $271$ | $150$ | $1^{15}\cdot2^{22}\cdot3^{14}\cdot4^{7}\cdot6$ |
68.7344.271-68.n.1.4 | $68$ | $2$ | $2$ | $271$ | $111$ | $1^{15}\cdot2^{22}\cdot3^{14}\cdot4^{7}\cdot6$ |
68.7344.271-68.o.1.4 | $68$ | $2$ | $2$ | $271$ | $115$ | $1^{15}\cdot2^{22}\cdot3^{14}\cdot4^{7}\cdot6$ |
68.7344.280-68.i.1.4 | $68$ | $2$ | $2$ | $280$ | $120$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
68.7344.280-68.k.1.4 | $68$ | $2$ | $2$ | $280$ | $113$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
68.7344.280-68.k.1.7 | $68$ | $2$ | $2$ | $280$ | $113$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
68.7344.280-68.l.1.6 | $68$ | $2$ | $2$ | $280$ | $122$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
68.7344.280-68.l.1.7 | $68$ | $2$ | $2$ | $280$ | $122$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
68.7344.280-68.n.1.5 | $68$ | $2$ | $2$ | $280$ | $115$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
68.7344.280-68.n.1.6 | $68$ | $2$ | $2$ | $280$ | $115$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |