Invariants
| Level: | $35$ | $\SL_2$-level: | $35$ | Newform level: | $1225$ | ||
| Index: | $2520$ | $\PSL_2$-index: | $1260$ | ||||
| Genus: | $88 = 1 + \frac{ 1260 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 36 }{2}$ | ||||||
| Cusps: | $36$ (none of which are rational) | Cusp widths | $35^{36}$ | Cusp orbits | $3^{2}\cdot6\cdot12^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $17$ | ||||||
| $\Q$-gonality: | $13 \le \gamma \le 21$ | ||||||
| $\overline{\Q}$-gonality: | $13 \le \gamma \le 21$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 35.2520.88.1 |
Level structure
| $\GL_2(\Z/35\Z)$-generators: | $\begin{bmatrix}6&30\\15&1\end{bmatrix}$, $\begin{bmatrix}17&11\\5&31\end{bmatrix}$, $\begin{bmatrix}22&1\\25&6\end{bmatrix}$ |
| $\GL_2(\Z/35\Z)$-subgroup: | $C_{12}\times \SD_{32}$ |
| Contains $-I$: | no $\quad$ (see 35.1260.88.a.1 for the level structure with $-I$) |
| Cyclic 35-isogeny field degree: | $8$ |
| Cyclic 35-torsion field degree: | $48$ |
| Full 35-torsion field degree: | $384$ |
Jacobian
| Conductor: | $5^{144}\cdot7^{176}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{2}\cdot2^{13}\cdot3^{2}\cdot4^{10}\cdot6\cdot8$ |
| Newforms: | 245.2.a.e$^{2}$, 245.2.a.f$^{2}$, 245.2.a.h$^{2}$, 245.2.b.c$^{2}$, 245.2.b.e$^{2}$, 245.2.b.f$^{2}$, 1225.2.a.ba, 1225.2.a.bb, 1225.2.a.bc, 1225.2.a.d, 1225.2.a.f, 1225.2.a.k, 1225.2.a.l, 1225.2.a.p, 1225.2.a.r, 1225.2.a.v, 1225.2.a.x, 1225.2.a.y, 1225.2.b.g, 1225.2.b.i, 1225.2.b.j, 1225.2.b.l, 1225.2.b.n |
Rational points
This modular curve has no $\Q_p$ points for $p=11,151,191$, and therefore no rational points.
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(5)$ | $5$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $120$ | $60$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(5)$ | $5$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 35.504.16-35.a.1.2 | $35$ | $5$ | $5$ | $16$ | $2$ | $1^{2}\cdot2^{9}\cdot3^{2}\cdot4^{8}\cdot6\cdot8$ |
| 35.504.16-35.a.2.2 | $35$ | $5$ | $5$ | $16$ | $2$ | $1^{2}\cdot2^{9}\cdot3^{2}\cdot4^{8}\cdot6\cdot8$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 35.5040.175-35.a.1.3 | $35$ | $2$ | $2$ | $175$ | $34$ | $1^{17}\cdot2^{21}\cdot3^{2}\cdot4^{4}\cdot6$ |
| 35.5040.175-35.b.1.4 | $35$ | $2$ | $2$ | $175$ | $30$ | $1^{17}\cdot2^{21}\cdot3^{2}\cdot4^{4}\cdot6$ |
| 35.5040.175-35.c.1.4 | $35$ | $2$ | $2$ | $175$ | $32$ | $1^{17}\cdot2^{21}\cdot3^{2}\cdot4^{4}\cdot6$ |
| 35.5040.175-35.d.1.3 | $35$ | $2$ | $2$ | $175$ | $32$ | $1^{17}\cdot2^{21}\cdot3^{2}\cdot4^{4}\cdot6$ |
| 70.5040.193-70.p.1.3 | $70$ | $2$ | $2$ | $193$ | $36$ | $1^{13}\cdot2^{14}\cdot3^{2}\cdot4^{11}\cdot6\cdot8$ |
| 70.5040.193-70.cj.1.3 | $70$ | $2$ | $2$ | $193$ | $41$ | $1^{13}\cdot2^{14}\cdot3^{2}\cdot4^{11}\cdot6\cdot8$ |
| 70.5040.193-70.ck.1.3 | $70$ | $2$ | $2$ | $193$ | $48$ | $1^{37}\cdot2^{16}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 70.5040.193-70.cl.1.2 | $70$ | $2$ | $2$ | $193$ | $37$ | $1^{37}\cdot2^{16}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 70.5040.193-70.co.1.3 | $70$ | $2$ | $2$ | $193$ | $35$ | $1^{37}\cdot2^{16}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 70.5040.193-70.cp.1.3 | $70$ | $2$ | $2$ | $193$ | $43$ | $1^{37}\cdot2^{16}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 70.5040.193-70.cq.1.2 | $70$ | $2$ | $2$ | $193$ | $42$ | $1^{13}\cdot2^{14}\cdot3^{2}\cdot4^{11}\cdot6\cdot8$ |
| 70.5040.193-70.cr.1.3 | $70$ | $2$ | $2$ | $193$ | $34$ | $1^{13}\cdot2^{14}\cdot3^{2}\cdot4^{11}\cdot6\cdot8$ |
| 70.7560.280-70.a.1.4 | $70$ | $3$ | $3$ | $280$ | $50$ | $1^{24}\cdot2^{36}\cdot3^{2}\cdot4^{17}\cdot6\cdot8^{2}$ |