Modular curve $X_{S_4}(13)$

• Label: 13.91.3.a.1
• Level: $13$
• Index: $91$
• Genus: $3$
• Analytic rank: $3$
• Cusps: $7$
• $\Q$-cusps: $0$

Invariants

Level:	$13$		$\SL_2$-level:	$13$		Newform level:	$169$
Index:	$91$		$\PSL_2$-index:	$91$
Genus:	$3 = 1 + \frac{ 91 }{12} - \frac{ 3 }{4} - \frac{ 4 }{3} - \frac{ 7 }{2}$
Cusps:	$7$ (none of which are rational)		Cusp widths	$13^{7}$		Cusp orbits	$3\cdot4$
Elliptic points:	$3$ of order $2$ and $4$ of order $3$
Analytic rank:	$3$
$\Q$-gonality:	$3$
$\overline{\Q}$-gonality:	$3$
Rational cusps:	$0$
Rational CM points:	yes $\quad(D =$ $-3$)

Other labels

Cummins and Pauli (CP) label:	13B3
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	13.91.3.2
Sutherland (S) label:	13S4

Level structure

$\GL_2(\Z/13\Z)$-generators:	$\begin{bmatrix}3&2\\11&5\end{bmatrix}$, $\begin{bmatrix}11&7\\1&2\end{bmatrix}$
$\GL_2(\Z/13\Z)$-subgroup:	$\SL(2,3):C_{12}$
Contains $-I$:	yes
Quadratic refinements:	none in database
Cyclic 13-isogeny field degree:	$6$
Cyclic 13-torsion field degree:	$72$
Full 13-torsion field degree:	$288$

Jacobian

Conductor:	$13^{6}$
Simple:	yes
Squarefree:	yes
Decomposition:	$3$
Newforms:	169.2.a.b

Models

Canonical model in $\mathbb{P}^{ 2 }$

$ 0 $

$=$

$ 5 x^{4} - 7 x^{3} y + 8 x^{3} z + 3 x^{2} y^{2} - 7 x^{2} y z + 4 x^{2} z^{2} + 2 x y^{3} + \cdots + 2 y z^{3} $

Rational points

This modular curve has rational points, including 1 rational CM point and 2 known non-cuspidal non-CM points, but no rational cusps. The following are the known rational points on this modular curve (one row per $j$-invariant).

Elliptic curve	CM	$j$-invariant		$j$-height	Canonical model
27.a3	$-3$	$0$		$0.000$	$(-1:-1:1)$
50700.j1	no	$\tfrac{11225615440}{1594323}$	$= 2^{4} \cdot 3^{-13} \cdot 5 \cdot 13^{4} \cdot 17^{3}$	$23.141$	$(0:1:0)$
61347.b1	no	$\tfrac{-160855552000}{1594323}$	$= -1 \cdot 2^{12} \cdot 3^{-13} \cdot 5^{3} \cdot 11 \cdot 13^{4}$	$25.804$	$(0:0:1)$

Maps to other modular curves

$j$-invariant map of degree 91 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 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Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X(1)$	$1$	$91$	$91$	$0$	$0$	full Jacobian

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
13.273.11.a.1	$13$	$3$	$3$	$11$	$6$	$2\cdot3^{2}$
13.364.16.b.1	$13$	$4$	$4$	$16$	$9$	$2^{2}\cdot3^{3}$
26.182.10.a.1	$26$	$2$	$2$	$10$	$3$	$1^{4}\cdot3$
26.182.10.b.1	$26$	$2$	$2$	$10$	$6$	$1^{4}\cdot3$
26.273.16.a.1	$26$	$3$	$3$	$16$	$8$	$1^{7}\cdot3^{2}$
39.273.18.a.1	$39$	$3$	$3$	$18$	$12$	$1\cdot2^{2}\cdot4\cdot6$
39.364.23.a.1	$39$	$4$	$4$	$23$	$11$	$1^{3}\cdot2^{4}\cdot3^{3}$
52.182.10.a.1	$52$	$2$	$2$	$10$	$3$	$1^{4}\cdot3$
52.182.10.b.1	$52$	$2$	$2$	$10$	$6$	$1^{4}\cdot3$
52.364.25.a.1	$52$	$4$	$4$	$25$	$19$	$1^{3}\cdot2^{5}\cdot3\cdot6$
65.455.32.a.1	$65$	$5$	$5$	$32$	$25$	$1^{8}\cdot2^{2}\cdot5\cdot12$
65.546.38.a.1	$65$	$6$	$6$	$38$	$17$	$1^{2}\cdot2^{6}\cdot3^{4}\cdot9$
65.910.67.a.1	$65$	$10$	$10$	$67$	$51$	$1^{10}\cdot2^{8}\cdot3^{4}\cdot5\cdot9\cdot12$
78.182.10.a.1	$78$	$2$	$2$	$10$	$?$	not computed
78.182.10.b.1	$78$	$2$	$2$	$10$	$?$	not computed
104.182.10.a.1	$104$	$2$	$2$	$10$	$?$	not computed
104.182.10.b.1	$104$	$2$	$2$	$10$	$?$	not computed
104.182.10.c.1	$104$	$2$	$2$	$10$	$?$	not computed
104.182.10.d.1	$104$	$2$	$2$	$10$	$?$	not computed
130.182.10.a.1	$130$	$2$	$2$	$10$	$?$	not computed
130.182.10.b.1	$130$	$2$	$2$	$10$	$?$	not computed
156.182.10.a.1	$156$	$2$	$2$	$10$	$?$	not computed
156.182.10.b.1	$156$	$2$	$2$	$10$	$?$	not computed
169.199927.16043.a.1	$169$	$2197$	$2197$	$16043$	$?$	not computed
182.182.10.a.1	$182$	$2$	$2$	$10$	$?$	not computed
182.182.10.b.1	$182$	$2$	$2$	$10$	$?$	not computed
260.182.10.a.1	$260$	$2$	$2$	$10$	$?$	not computed
260.182.10.b.1	$260$	$2$	$2$	$10$	$?$	not computed
286.182.10.a.1	$286$	$2$	$2$	$10$	$?$	not computed
286.182.10.b.1	$286$	$2$	$2$	$10$	$?$	not computed
312.182.10.a.1	$312$	$2$	$2$	$10$	$?$	not computed
312.182.10.b.1	$312$	$2$	$2$	$10$	$?$	not computed
312.182.10.c.1	$312$	$2$	$2$	$10$	$?$	not computed
312.182.10.d.1	$312$	$2$	$2$	$10$	$?$	not computed