Modular curve 65.9360.355-65.a.1.3

• Label: 65.9360.355-65.a.1.3
• Level: $65$
• Index: $9360$
• Genus: $355$
• Analytic rank: $83$
• Cusps: $72$
• $\Q$-cusps: $0$

Invariants

Level:	$65$		$\SL_2$-level:	$65$		Newform level:	$4225$
Index:	$9360$		$\PSL_2$-index:	$4680$
Genus:	$355 = 1 + \frac{ 4680 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 72 }{2}$
Cusps:	$72$ (none of which are rational)		Cusp widths	$65^{72}$		Cusp orbits	$6^{2}\cdot12\cdot24^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$83$
$\Q$-gonality:	$47 \le \gamma \le 78$
$\overline{\Q}$-gonality:	$47 \le \gamma \le 78$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label:

65.9360.355.1

Level structure

$\GL_2(\Z/65\Z)$-generators:	$\begin{bmatrix}21&25\\10&57\end{bmatrix}$, $\begin{bmatrix}21&30\\35&57\end{bmatrix}$, $\begin{bmatrix}56&25\\35&61\end{bmatrix}$
$\GL_2(\Z/65\Z)$-subgroup:	$C_{84}:\OD_{16}$
Contains $-I$:	no $\quad$ (see 65.4680.355.a.1 for the level structure with $-I$)
Cyclic 65-isogeny field degree:	$14$
Cyclic 65-torsion field degree:	$168$
Full 65-torsion field degree:	$1344$

Jacobian

Conductor:	$5^{570}\cdot13^{710}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{7}\cdot2^{23}\cdot3^{9}\cdot4^{7}\cdot5^{4}\cdot6^{10}\cdot9^{3}\cdot10^{2}\cdot12^{2}\cdot18^{4}\cdot24$
Newforms:	169.2.a.b$^{3}$, 845.2.a.a$^{2}$, 845.2.a.b$^{2}$, 845.2.a.c$^{2}$, 845.2.a.d$^{2}$, 845.2.a.e$^{2}$, 845.2.a.f$^{2}$, 845.2.a.g$^{2}$, 845.2.a.j$^{2}$, 845.2.a.o$^{2}$, 845.2.b.c$^{2}$, 845.2.b.d$^{2}$, 845.2.b.e$^{2}$, 845.2.b.h$^{2}$, 4225.2.a.b, 4225.2.a.ba, 4225.2.a.bd, 4225.2.a.bg, 4225.2.a.bh, 4225.2.a.bm, 4225.2.a.bn, 4225.2.a.bo, 4225.2.a.bp, 4225.2.a.bq, 4225.2.a.br, 4225.2.a.bs, 4225.2.a.bx, 4225.2.a.by, 4225.2.a.cb, 4225.2.a.g, 4225.2.a.i, 4225.2.a.j, 4225.2.a.p, 4225.2.a.r, 4225.2.a.s, 4225.2.a.t, 4225.2.a.u, 4225.2.a.w, 4225.2.a.x, 4225.2.a.y, 4225.2.a.z, 4225.2.b.a, 4225.2.b.bc, 4225.2.b.d, 4225.2.b.g, 4225.2.b.h, 4225.2.b.i, 4225.2.b.j, 4225.2.b.l, 4225.2.b.m, 4225.2.b.n, 4225.2.b.o, 4225.2.b.r, 4225.2.b.t, 4225.2.b.x, 4225.2.b.y, 4225.2.b.z

Rational points

This modular curve has no $\Q_p$ points for $p=2,11,31,41,101,241,331,401$, 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
$X_{\mathrm{arith}}(5)$	$5$	$78$	$78$	$0$	$0$	full Jacobian
$X_{\mathrm{ns}}^+(13)$	$13$	$120$	$60$	$3$	$3$	$1^{7}\cdot2^{23}\cdot3^{8}\cdot4^{7}\cdot5^{4}\cdot6^{10}\cdot9^{3}\cdot10^{2}\cdot12^{2}\cdot18^{4}\cdot24$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X_{\mathrm{arith}}(5)$	$5$	$78$	$78$	$0$	$0$	full Jacobian
65.1872.67-65.a.1.3	$65$	$5$	$5$	$67$	$15$	$1^{6}\cdot2^{17}\cdot3^{6}\cdot4^{7}\cdot5^{4}\cdot6^{7}\cdot9^{2}\cdot10^{2}\cdot12^{2}\cdot18^{3}\cdot24$
65.1872.67-65.a.2.2	$65$	$5$	$5$	$67$	$15$	$1^{6}\cdot2^{17}\cdot3^{6}\cdot4^{7}\cdot5^{4}\cdot6^{7}\cdot9^{2}\cdot10^{2}\cdot12^{2}\cdot18^{3}\cdot24$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
65.18720.709-65.a.1.3	$65$	$2$	$2$	$709$	$159$	$1^{12}\cdot2^{12}\cdot3^{13}\cdot4^{9}\cdot6^{4}\cdot8^{4}\cdot9^{3}\cdot10^{2}\cdot12^{2}\cdot18^{4}\cdot20\cdot24$
65.18720.709-65.b.1.4	$65$	$2$	$2$	$709$	$165$	$1^{12}\cdot2^{12}\cdot3^{13}\cdot4^{9}\cdot6^{4}\cdot8^{4}\cdot9^{3}\cdot10^{2}\cdot12^{2}\cdot18^{4}\cdot20\cdot24$
65.18720.709-65.c.1.4	$65$	$2$	$2$	$709$	$167$	$1^{12}\cdot2^{12}\cdot3^{13}\cdot4^{9}\cdot6^{4}\cdot8^{4}\cdot9^{3}\cdot10^{2}\cdot12^{2}\cdot18^{4}\cdot20\cdot24$
65.18720.709-65.d.1.4	$65$	$2$	$2$	$709$	$149$	$1^{12}\cdot2^{12}\cdot3^{13}\cdot4^{9}\cdot6^{4}\cdot8^{4}\cdot9^{3}\cdot10^{2}\cdot12^{2}\cdot18^{4}\cdot20\cdot24$
65.65520.2479-65.c.1.5	$65$	$7$	$7$	$2479$	$551$	$1^{78}\cdot2^{138}\cdot3^{65}\cdot4^{57}\cdot5^{12}\cdot6^{46}\cdot8^{12}\cdot9^{15}\cdot10^{12}\cdot12^{10}\cdot18^{20}\cdot20^{3}\cdot24^{5}$