Properties

Label 65.10920.414-65.a.1.4
Level $65$
Index $10920$
Genus $414$
Analytic rank $93$
Cusps $84$
$\Q$-cusps $0$

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Invariants

Level: $65$ $\SL_2$-level: $65$ Newform level: $4225$
Index: $10920$ $\PSL_2$-index:$5460$
Genus: $414 = 1 + \frac{ 5460 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 84 }{2}$
Cusps: $84$ (none of which are rational) Cusp widths $65^{84}$ Cusp orbits $3^{2}\cdot4^{2}\cdot6\cdot8\cdot12^{2}\cdot16^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $93$
$\Q$-gonality: $55 \le \gamma \le 91$
$\overline{\Q}$-gonality: $55 \le \gamma \le 91$
Rational cusps: $0$
Rational CM points: none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label: 65.10920.414.2

Level structure

$\GL_2(\Z/65\Z)$-generators: $\begin{bmatrix}46&10\\30&27\end{bmatrix}$, $\begin{bmatrix}46&20\\30&34\end{bmatrix}$
Contains $-I$: no $\quad$ (see 65.5460.414.a.1 for the level structure with $-I$)
Cyclic 65-isogeny field degree: $6$
Cyclic 65-torsion field degree: $72$
Full 65-torsion field degree: $1152$

Jacobian

Conductor: $5^{666}\cdot13^{769}$
Simple: no
Squarefree: no
Decomposition: $1^{26}\cdot2^{36}\cdot3^{17}\cdot4^{8}\cdot5^{2}\cdot6^{11}\cdot9^{3}\cdot10\cdot12^{2}\cdot18^{4}\cdot24$
Newforms: 65.2.a.a$^{2}$, 65.2.a.b$^{2}$, 65.2.a.c$^{2}$, 65.2.b.a$^{2}$, 169.2.a.b$^{3}$, 325.2.a.a, 325.2.a.b, 325.2.a.c, 325.2.a.d, 325.2.a.e, 325.2.a.f, 325.2.a.g, 325.2.a.h, 325.2.a.i, 325.2.a.j, 325.2.a.k, 325.2.b.a, 325.2.b.b, 325.2.b.c, 325.2.b.d, 325.2.b.e, 325.2.b.f, 845.2.a.a$^{2}$, 845.2.a.c$^{2}$, 845.2.a.d$^{2}$, 845.2.a.e$^{2}$, 845.2.a.g$^{2}$, 845.2.a.i$^{2}$, 845.2.a.j$^{2}$, 845.2.a.k$^{2}$, 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.h$^{2}$, 4225.2.a.a, 4225.2.a.b, 4225.2.a.ba, 4225.2.a.bc, 4225.2.a.bd, 4225.2.a.be, 4225.2.a.bg, 4225.2.a.bh, 4225.2.a.bn, 4225.2.a.bp, 4225.2.a.br, 4225.2.a.bs, 4225.2.a.bx, 4225.2.a.by, 4225.2.a.c, 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.u, 4225.2.a.w, 4225.2.a.x, 4225.2.a.z, 4225.2.b.a, 4225.2.b.b, 4225.2.b.bc, 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.j, 4225.2.b.l, 4225.2.b.m, 4225.2.b.n, 4225.2.b.p, 4225.2.b.q, 4225.2.b.r, 4225.2.b.t, 4225.2.b.y, 4225.2.b.z

Rational points

This modular curve has no $\Q_p$ points for $p=7,11,41,\ldots,1931$, 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
$X_{\mathrm{arith}}(5)$ $5$ $91$ $91$ $0$ $0$ full Jacobian
$X_{S_4}(13)$ $13$ $120$ $60$ $3$ $3$ $1^{26}\cdot2^{36}\cdot3^{16}\cdot4^{8}\cdot5^{2}\cdot6^{11}\cdot9^{3}\cdot10\cdot12^{2}\cdot18^{4}\cdot24$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_{\mathrm{arith}}(5)$ $5$ $91$ $91$ $0$ $0$ full Jacobian
65.2184.78-65.a.1.2 $65$ $5$ $5$ $78$ $17$ $1^{24}\cdot2^{28}\cdot3^{12}\cdot4^{8}\cdot5^{2}\cdot6^{8}\cdot9^{2}\cdot10\cdot12^{2}\cdot18^{3}\cdot24$
65.2184.78-65.a.2.1 $65$ $5$ $5$ $78$ $17$ $1^{24}\cdot2^{28}\cdot3^{12}\cdot4^{8}\cdot5^{2}\cdot6^{8}\cdot9^{2}\cdot10\cdot12^{2}\cdot18^{3}\cdot24$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
65.32760.1240-65.a.1.6 $65$ $3$ $3$ $1240$ $273$ $1^{14}\cdot2^{50}\cdot3^{18}\cdot4^{23}\cdot5^{8}\cdot6^{20}\cdot8^{4}\cdot9^{6}\cdot10^{6}\cdot12^{4}\cdot18^{8}\cdot20\cdot24^{2}$
65.43680.1653-65.b.1.3 $65$ $4$ $4$ $1653$ $371$ $1^{45}\cdot2^{75}\cdot3^{39}\cdot4^{33}\cdot5^{6}\cdot6^{25}\cdot8^{8}\cdot9^{9}\cdot10^{7}\cdot12^{6}\cdot18^{12}\cdot20^{2}\cdot24^{3}$