Modular curve 65.18720.709-65.d.1.4

• Label: 65.18720.709-65.d.1.4
• Level: $65$
• Index: $18720$
• Genus: $709$
• Analytic rank: $149$
• Cusps: $144$
• $\Q$-cusps: $0$

Invariants

Level:	$65$		$\SL_2$-level:	$65$		Newform level:	$4225$
Index:	$18720$		$\PSL_2$-index:	$9360$
Genus:	$709 = 1 + \frac{ 9360 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 144 }{2}$
Cusps:	$144$ (none of which are rational)		Cusp widths	$65^{144}$		Cusp orbits	$12^{2}\cdot24\cdot48^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$149$
$\Q$-gonality:	$93 \le \gamma \le 156$
$\overline{\Q}$-gonality:	$93 \le \gamma \le 156$
Rational cusps:	$0$
Rational CM points:	none

Other labels

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

65.18720.709.3

Level structure

$\GL_2(\Z/65\Z)$-generators:	$\begin{bmatrix}26&40\\5&13\end{bmatrix}$, $\begin{bmatrix}56&60\\25&19\end{bmatrix}$
$\GL_2(\Z/65\Z)$-subgroup:	$C_{168}:C_4$
Contains $-I$:	no $\quad$ (see 65.9360.709.d.1 for the level structure with $-I$)
Cyclic 65-isogeny field degree:	$14$
Cyclic 65-torsion field degree:	$168$
Full 65-torsion field degree:	$672$

Jacobian

Conductor:	$5^{1182}\cdot13^{1418}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{19}\cdot2^{35}\cdot3^{22}\cdot4^{16}\cdot5^{4}\cdot6^{14}\cdot8^{4}\cdot9^{6}\cdot10^{4}\cdot12^{4}\cdot18^{8}\cdot20\cdot24^{2}$
Newforms:	169.2.a.a, 169.2.a.b$^{3}$, 169.2.a.c, 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.h, 845.2.a.i, 845.2.a.j$^{2}$, 845.2.a.k, 845.2.a.l, 845.2.a.m, 845.2.a.n, 845.2.a.o$^{2}$, 845.2.b.a$^{2}$, 845.2.b.b$^{2}$, 845.2.b.c$^{2}$, 845.2.b.d$^{2}$, 845.2.b.e$^{2}$, 845.2.b.f$^{2}$, 845.2.b.g$^{2}$, 845.2.b.h$^{2}$, 4225.2.a.a, 4225.2.a.b, 4225.2.a.ba, 4225.2.a.bb$^{3}$, 4225.2.a.bc$^{2}$, 4225.2.a.bd, 4225.2.a.be$^{2}$, 4225.2.a.bf$^{2}$, 4225.2.a.bg, 4225.2.a.bh, 4225.2.a.bi$^{2}$, 4225.2.a.bj, 4225.2.a.bk, 4225.2.a.bl$^{2}$, 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.bt$^{2}$, 4225.2.a.bu, 4225.2.a.bv, 4225.2.a.bw, 4225.2.a.bx, 4225.2.a.by, 4225.2.a.bz, 4225.2.a.c, 4225.2.a.ca, 4225.2.a.cb, 4225.2.a.d, 4225.2.a.e, 4225.2.a.f, 4225.2.a.g, 4225.2.a.h, 4225.2.a.i, 4225.2.a.j, 4225.2.a.k, 4225.2.a.l, 4225.2.a.m, 4225.2.a.n, 4225.2.a.o, 4225.2.a.p, 4225.2.a.q, 4225.2.a.r, 4225.2.a.s, 4225.2.a.t, 4225.2.a.u, 4225.2.a.v$^{3}$, 4225.2.a.w, 4225.2.a.x, 4225.2.a.y, 4225.2.a.z, 4225.2.b.a, 4225.2.b.b, 4225.2.b.ba, 4225.2.b.bb, 4225.2.b.bc, 4225.2.b.bd, 4225.2.b.c, 4225.2.b.d, 4225.2.b.e, 4225.2.b.f, 4225.2.b.g, 4225.2.b.h, 4225.2.b.i, 4225.2.b.j, 4225.2.b.k, 4225.2.b.l, 4225.2.b.m, 4225.2.b.n, 4225.2.b.o, 4225.2.b.p, 4225.2.b.q, 4225.2.b.r, 4225.2.b.s, 4225.2.b.t, 4225.2.b.u, 4225.2.b.v, 4225.2.b.w, 4225.2.b.x, 4225.2.b.y, 4225.2.b.z

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=2,3,\ldots,401$, and therefore no rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
65.3744.133-65.b.1.4	$65$	$5$	$5$	$133$	$15$	$1^{18}\cdot2^{25}\cdot3^{14}\cdot4^{14}\cdot5^{4}\cdot6^{11}\cdot8^{3}\cdot9^{4}\cdot10^{4}\cdot12^{4}\cdot18^{6}\cdot20\cdot24^{2}$
65.3744.133-65.b.2.3	$65$	$5$	$5$	$133$	$15$	$1^{18}\cdot2^{25}\cdot3^{14}\cdot4^{14}\cdot5^{4}\cdot6^{11}\cdot8^{3}\cdot9^{4}\cdot10^{4}\cdot12^{4}\cdot18^{6}\cdot20\cdot24^{2}$
65.9360.355-65.a.1.3	$65$	$2$	$2$	$355$	$83$	$1^{12}\cdot2^{12}\cdot3^{13}\cdot4^{9}\cdot6^{4}\cdot8^{4}\cdot9^{3}\cdot10^{2}\cdot12^{2}\cdot18^{4}\cdot20\cdot24$
65.9360.355-65.a.1.5	$65$	$2$	$2$	$355$	$83$	$1^{12}\cdot2^{12}\cdot3^{13}\cdot4^{9}\cdot6^{4}\cdot8^{4}\cdot9^{3}\cdot10^{2}\cdot12^{2}\cdot18^{4}\cdot20\cdot24$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
65.131040.4957-65.h.1.4	$65$	$7$	$7$	$4957$	$1109$	$1^{156}\cdot2^{276}\cdot3^{130}\cdot4^{114}\cdot5^{24}\cdot6^{92}\cdot8^{24}\cdot9^{30}\cdot10^{24}\cdot12^{20}\cdot18^{40}\cdot20^{6}\cdot24^{10}$