Invariants
| Level: | $35$ | $\SL_2$-level: | $35$ | Newform level: | $1225$ | ||
| Index: | $3360$ | $\PSL_2$-index: | $1680$ | ||||
| Genus: | $117 = 1 + \frac{ 1680 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 48 }{2}$ | ||||||
| Cusps: | $48$ (of which $2$ are rational) | Cusp widths | $35^{48}$ | Cusp orbits | $1^{2}\cdot2\cdot3^{2}\cdot4^{2}\cdot6\cdot12^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $19$ | ||||||
| $\Q$-gonality: | $17 \le \gamma \le 28$ | ||||||
| $\overline{\Q}$-gonality: | $17 \le \gamma \le 28$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 35.3360.117.1 |
Level structure
| $\GL_2(\Z/35\Z)$-generators: | $\begin{bmatrix}14&12\\30&21\end{bmatrix}$, $\begin{bmatrix}26&5\\5&16\end{bmatrix}$, $\begin{bmatrix}28&18\\25&1\end{bmatrix}$ |
| $\GL_2(\Z/35\Z)$-subgroup: | $C_6^2.C_2^3$ |
| Contains $-I$: | no $\quad$ (see 35.1680.117.a.1 for the level structure with $-I$) |
| Cyclic 35-isogeny field degree: | $2$ |
| Cyclic 35-torsion field degree: | $12$ |
| Full 35-torsion field degree: | $288$ |
Jacobian
| Conductor: | $5^{192}\cdot7^{205}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}\cdot2^{21}\cdot3^{2}\cdot4^{12}\cdot6\cdot8$ |
| Newforms: | 35.2.a.a$^{2}$, 35.2.a.b$^{2}$, 35.2.b.a$^{2}$, 175.2.a.a, 175.2.a.b, 175.2.a.c, 175.2.a.d, 175.2.a.e, 175.2.a.f, 175.2.b.a, 175.2.b.b, 175.2.b.c, 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 2 rational cusps but no known non-cuspidal 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(5)$ | $5$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $120$ | $60$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(5)$ | $5$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| 35.672.21-35.a.1.6 | $35$ | $5$ | $5$ | $21$ | $2$ | $1^{6}\cdot2^{15}\cdot3^{2}\cdot4^{10}\cdot6\cdot8$ |
| 35.672.21-35.a.2.2 | $35$ | $5$ | $5$ | $21$ | $2$ | $1^{6}\cdot2^{15}\cdot3^{2}\cdot4^{10}\cdot6\cdot8$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 35.6720.233-35.a.1.5 | $35$ | $2$ | $2$ | $233$ | $38$ | $1^{22}\cdot2^{29}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 35.6720.233-35.b.1.5 | $35$ | $2$ | $2$ | $233$ | $38$ | $1^{22}\cdot2^{29}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 35.6720.233-35.c.1.4 | $35$ | $2$ | $2$ | $233$ | $39$ | $1^{22}\cdot2^{29}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 35.6720.233-35.d.1.4 | $35$ | $2$ | $2$ | $233$ | $37$ | $1^{22}\cdot2^{29}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 35.10080.349-35.b.1.6 | $35$ | $3$ | $3$ | $349$ | $60$ | $1^{26}\cdot2^{47}\cdot3^{6}\cdot4^{17}\cdot6^{3}\cdot8$ |
| 70.6720.257-70.m.1.4 | $70$ | $2$ | $2$ | $257$ | $43$ | $1^{30}\cdot2^{23}\cdot3^{2}\cdot4^{11}\cdot6\cdot8$ |
| 70.6720.257-70.ck.1.6 | $70$ | $2$ | $2$ | $257$ | $52$ | $1^{30}\cdot2^{23}\cdot3^{2}\cdot4^{11}\cdot6\cdot8$ |
| 70.6720.257-70.cl.1.2 | $70$ | $2$ | $2$ | $257$ | $56$ | $1^{54}\cdot2^{25}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 70.6720.257-70.cm.1.1 | $70$ | $2$ | $2$ | $257$ | $50$ | $1^{54}\cdot2^{25}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 70.6720.257-70.cp.1.6 | $70$ | $2$ | $2$ | $257$ | $46$ | $1^{54}\cdot2^{25}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 70.6720.257-70.cq.1.6 | $70$ | $2$ | $2$ | $257$ | $50$ | $1^{54}\cdot2^{25}\cdot3^{2}\cdot4^{6}\cdot6$ |
| 70.6720.257-70.cr.1.4 | $70$ | $2$ | $2$ | $257$ | $55$ | $1^{30}\cdot2^{23}\cdot3^{2}\cdot4^{11}\cdot6\cdot8$ |
| 70.6720.257-70.cs.1.4 | $70$ | $2$ | $2$ | $257$ | $42$ | $1^{30}\cdot2^{23}\cdot3^{2}\cdot4^{11}\cdot6\cdot8$ |
| 70.10080.373-70.a.1.6 | $70$ | $3$ | $3$ | $373$ | $56$ | $1^{42}\cdot2^{51}\cdot3^{2}\cdot4^{21}\cdot6\cdot8^{2}$ |