Modular curve 65.168.11.a.2

• Label: 65.168.11.a.2
• Level: $65$
• Index: $168$
• Genus: $11$
• Analytic rank: $1$
• Cusps: $8$
• $\Q$-cusps: $4$

Invariants

Level:	$65$		$\SL_2$-level:	$65$		Newform level:	$65$
Index:	$168$		$\PSL_2$-index:	$168$
Genus:	$11 = 1 + \frac{ 168 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (of which $4$ are rational)		Cusp widths	$1^{2}\cdot5^{2}\cdot13^{2}\cdot65^{2}$		Cusp orbits	$1^{4}\cdot2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$3 \le \gamma \le 10$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 10$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	65A11
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	65.168.11.2

Level structure

$\GL_2(\Z/65\Z)$-generators:	$\begin{bmatrix}18&50\\0&36\end{bmatrix}$, $\begin{bmatrix}27&19\\0&51\end{bmatrix}$, $\begin{bmatrix}51&2\\0&11\end{bmatrix}$, $\begin{bmatrix}58&29\\0&4\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	65.336.11-65.a.2.1, 65.336.11-65.a.2.2, 65.336.11-65.a.2.3, 65.336.11-65.a.2.4, 65.336.11-65.a.2.5, 65.336.11-65.a.2.6, 65.336.11-65.a.2.7, 65.336.11-65.a.2.8, 130.336.11-65.a.2.1, 130.336.11-65.a.2.2, 130.336.11-65.a.2.3, 130.336.11-65.a.2.4, 130.336.11-65.a.2.5, 130.336.11-65.a.2.6, 130.336.11-65.a.2.7, 130.336.11-65.a.2.8, 195.336.11-65.a.2.1, 195.336.11-65.a.2.2, 195.336.11-65.a.2.3, 195.336.11-65.a.2.4, 195.336.11-65.a.2.5, 195.336.11-65.a.2.6, 195.336.11-65.a.2.7, 195.336.11-65.a.2.8, 260.336.11-65.a.2.1, 260.336.11-65.a.2.2, 260.336.11-65.a.2.3, 260.336.11-65.a.2.4, 260.336.11-65.a.2.5, 260.336.11-65.a.2.6, 260.336.11-65.a.2.7, 260.336.11-65.a.2.8, 260.336.11-65.a.2.9, 260.336.11-65.a.2.10, 260.336.11-65.a.2.11, 260.336.11-65.a.2.12, 260.336.11-65.a.2.13, 260.336.11-65.a.2.14, 260.336.11-65.a.2.15, 260.336.11-65.a.2.16
Cyclic 65-isogeny field degree:	$1$
Cyclic 65-torsion field degree:	$48$
Full 65-torsion field degree:	$74880$

Jacobian

Conductor:	$5^{11}\cdot13^{11}$
Simple:	no
Squarefree:	yes
Decomposition:	$1\cdot2^{2}\cdot6$
Newforms:	65.2.a.a, 65.2.a.b, 65.2.a.c, 65.2.b.a

Models

Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations

$ 0 $	$=$	$ x^{2} - x w + x t - x u - x v - x r + x s - x a - x b + v s + v a + s a + a^{2} $
	$=$	$x^{2} - x z + x v + x s + x a + y^{2} + 2 y t - z t + 2 v s + 2 v a + s a + s b + a^{2} + a b$
	$=$	$x^{2} - x y - x w + x t + x u + x a + y z + 2 y t + u s + u a + r s + r a - s^{2} + a^{2}$
	$=$	$x^{2} - x y + 2 x z + x w + 2 x t - 2 x v + x r + x a - x b - y^{2} + y z - y w - y s + z t + s^{2} + \cdots - a b$
	$=$	$\cdots$

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points.

Modular covers

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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
5.12.0.a.2	$5$	$14$	$14$	$0$	$0$	full Jacobian
$X_0(13)$	$13$	$12$	$12$	$0$	$0$	full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
5.12.0.a.2	$5$	$14$	$14$	$0$	$0$	full Jacobian
$X_0(65)$	$65$	$2$	$2$	$5$	$1$	$6$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
65.336.21.a.3	$65$	$2$	$2$	$21$	$1$	$2^{2}\cdot6$
65.336.21.a.4	$65$	$2$	$2$	$21$	$1$	$2^{2}\cdot6$
65.336.21.b.3	$65$	$2$	$2$	$21$	$1$	$2^{2}\cdot6$
65.336.21.b.4	$65$	$2$	$2$	$21$	$1$	$2^{2}\cdot6$
65.504.31.a.1	$65$	$3$	$3$	$31$	$1$	$4^{2}\cdot12$
65.504.31.a.3	$65$	$3$	$3$	$31$	$1$	$4^{2}\cdot12$
65.504.31.b.2	$65$	$3$	$3$	$31$	$5$	$2^{4}\cdot6^{2}$
65.840.59.a.1	$65$	$5$	$5$	$59$	$9$	$1^{6}\cdot2^{9}\cdot3^{2}\cdot4^{3}\cdot6$
65.2184.155.a.2	$65$	$13$	$13$	$155$	$27$	$1^{2}\cdot2^{12}\cdot3^{8}\cdot4^{2}\cdot6^{4}\cdot8\cdot9^{2}\cdot18^{2}$
195.336.21.g.2	$195$	$2$	$2$	$21$	$?$	not computed
195.336.21.g.4	$195$	$2$	$2$	$21$	$?$	not computed
195.336.21.j.3	$195$	$2$	$2$	$21$	$?$	not computed
195.336.21.j.4	$195$	$2$	$2$	$21$	$?$	not computed
260.336.21.gm.1	$260$	$2$	$2$	$21$	$?$	not computed
260.336.21.gm.2	$260$	$2$	$2$	$21$	$?$	not computed
260.336.21.gp.1	$260$	$2$	$2$	$21$	$?$	not computed
260.336.21.gp.2	$260$	$2$	$2$	$21$	$?$	not computed