Modular curve 4.6.0.b.1

• Label: 4.6.0.b.1
• Level: $4$
• Index: $6$
• Genus: $0$
• Analytic rank: $0$
• Cusps: $3$
• $\Q$-cusps: $1$

Invariants

Level:	$4$		$\SL_2$-level:	$4$
Index:	$6$		$\PSL_2$-index:	$6$
Genus:	$0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$
Cusps:	$3$ (of which $1$ is rational)		Cusp widths	$1^{2}\cdot4$		Cusp orbits	$1\cdot2$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
$\Q$-gonality:	$1$
$\overline{\Q}$-gonality:	$1$
Rational cusps:	$1$
Rational CM points:	yes $\quad(D =$ $-4$)

Other labels

Cummins and Pauli (CP) label:	4B0
Sutherland and Zywina (SZ) label:	4B0-4a
Rouse and Zureick-Brown (RZB) label:	X9
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	4.6.0.3

Level structure

$\GL_2(\Z/4\Z)$-generators:	$\begin{bmatrix}1&1\\2&1\end{bmatrix}$, $\begin{bmatrix}1&3\\0&1\end{bmatrix}$
$\GL_2(\Z/4\Z)$-subgroup:	$C_2^2:C_4$
Contains $-I$:	yes
Quadratic refinements:	8.12.0-4.b.1.1, 8.12.0-4.b.1.2, 8.12.0-4.b.1.3, 8.12.0-4.b.1.4, 24.12.0-4.b.1.1, 24.12.0-4.b.1.2, 24.12.0-4.b.1.3, 24.12.0-4.b.1.4, 40.12.0-4.b.1.1, 40.12.0-4.b.1.2, 40.12.0-4.b.1.3, 40.12.0-4.b.1.4, 56.12.0-4.b.1.1, 56.12.0-4.b.1.2, 56.12.0-4.b.1.3, 56.12.0-4.b.1.4, 88.12.0-4.b.1.1, 88.12.0-4.b.1.2, 88.12.0-4.b.1.3, 88.12.0-4.b.1.4, 104.12.0-4.b.1.1, 104.12.0-4.b.1.2, 104.12.0-4.b.1.3, 104.12.0-4.b.1.4, 120.12.0-4.b.1.1, 120.12.0-4.b.1.2, 120.12.0-4.b.1.3, 120.12.0-4.b.1.4, 136.12.0-4.b.1.1, 136.12.0-4.b.1.2, 136.12.0-4.b.1.3, 136.12.0-4.b.1.4, 152.12.0-4.b.1.1, 152.12.0-4.b.1.2, 152.12.0-4.b.1.3, 152.12.0-4.b.1.4, 168.12.0-4.b.1.1, 168.12.0-4.b.1.2, 168.12.0-4.b.1.3, 168.12.0-4.b.1.4, 184.12.0-4.b.1.1, 184.12.0-4.b.1.2, 184.12.0-4.b.1.3, 184.12.0-4.b.1.4, 232.12.0-4.b.1.1, 232.12.0-4.b.1.2, 232.12.0-4.b.1.3, 232.12.0-4.b.1.4, 248.12.0-4.b.1.1, 248.12.0-4.b.1.2, 248.12.0-4.b.1.3, 248.12.0-4.b.1.4, 264.12.0-4.b.1.1, 264.12.0-4.b.1.2, 264.12.0-4.b.1.3, 264.12.0-4.b.1.4, 280.12.0-4.b.1.1, 280.12.0-4.b.1.2, 280.12.0-4.b.1.3, 280.12.0-4.b.1.4, 296.12.0-4.b.1.1, 296.12.0-4.b.1.2, 296.12.0-4.b.1.3, 296.12.0-4.b.1.4, 312.12.0-4.b.1.1, 312.12.0-4.b.1.2, 312.12.0-4.b.1.3, 312.12.0-4.b.1.4, 328.12.0-4.b.1.1, 328.12.0-4.b.1.2, 328.12.0-4.b.1.3, 328.12.0-4.b.1.4
Cyclic 4-isogeny field degree:	$2$
Cyclic 4-torsion field degree:	$4$
Full 4-torsion field degree:	$16$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has infinitely many rational points, including 11629 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 6 to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{x^{6}(x^{2}+48y^{2})^{3}}{y^{4}x^{6}(x^{2}+64y^{2})}$

Modular covers

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
$X_0(2)$	$2$	$2$	$2$	$0$	$0$

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

Covering curve	Level	Index	Degree	Genus
4.12.0.a.1	$4$	$2$	$2$	$0$
4.12.0.c.1	$4$	$2$	$2$	$0$
8.12.0.c.1	$8$	$2$	$2$	$0$
8.12.0.h.1	$8$	$2$	$2$	$0$
12.12.0.e.1	$12$	$2$	$2$	$0$
12.12.0.f.1	$12$	$2$	$2$	$0$
12.18.1.b.1	$12$	$3$	$3$	$1$
12.24.0.f.1	$12$	$4$	$4$	$0$
20.12.0.e.1	$20$	$2$	$2$	$0$
20.12.0.f.1	$20$	$2$	$2$	$0$
20.30.2.b.1	$20$	$5$	$5$	$2$
20.36.1.b.1	$20$	$6$	$6$	$1$
20.60.3.b.1	$20$	$10$	$10$	$3$
24.12.0.m.1	$24$	$2$	$2$	$0$
24.12.0.p.1	$24$	$2$	$2$	$0$
28.12.0.e.1	$28$	$2$	$2$	$0$
28.12.0.f.1	$28$	$2$	$2$	$0$
28.48.2.b.1	$28$	$8$	$8$	$2$
28.126.7.b.1	$28$	$21$	$21$	$7$
28.168.9.b.1	$28$	$28$	$28$	$9$
36.162.10.c.1	$36$	$27$	$27$	$10$
40.12.0.m.1	$40$	$2$	$2$	$0$
40.12.0.p.1	$40$	$2$	$2$	$0$
44.12.0.e.1	$44$	$2$	$2$	$0$
44.12.0.f.1	$44$	$2$	$2$	$0$
44.72.4.b.1	$44$	$12$	$12$	$4$
44.330.21.c.1	$44$	$55$	$55$	$21$
44.330.21.e.1	$44$	$55$	$55$	$21$
44.396.25.b.1	$44$	$66$	$66$	$25$
52.12.0.e.1	$52$	$2$	$2$	$0$
52.12.0.f.1	$52$	$2$	$2$	$0$
52.84.5.b.1	$52$	$14$	$14$	$5$
52.468.31.b.1	$52$	$78$	$78$	$31$
52.546.36.c.1	$52$	$91$	$91$	$36$
52.546.36.e.1	$52$	$91$	$91$	$36$
56.12.0.m.1	$56$	$2$	$2$	$0$
56.12.0.p.1	$56$	$2$	$2$	$0$
60.12.0.e.1	$60$	$2$	$2$	$0$
60.12.0.f.1	$60$	$2$	$2$	$0$
68.12.0.e.1	$68$	$2$	$2$	$0$
68.12.0.f.1	$68$	$2$	$2$	$0$
68.108.7.b.1	$68$	$18$	$18$	$7$
68.816.57.b.1	$68$	$136$	$136$	$57$
68.918.64.b.1	$68$	$153$	$153$	$64$
76.12.0.e.1	$76$	$2$	$2$	$0$
76.12.0.f.1	$76$	$2$	$2$	$0$
76.120.8.b.1	$76$	$20$	$20$	$8$
84.12.0.e.1	$84$	$2$	$2$	$0$
84.12.0.f.1	$84$	$2$	$2$	$0$
88.12.0.m.1	$88$	$2$	$2$	$0$
88.12.0.p.1	$88$	$2$	$2$	$0$
92.12.0.e.1	$92$	$2$	$2$	$0$
92.12.0.f.1	$92$	$2$	$2$	$0$
92.144.10.b.1	$92$	$24$	$24$	$10$
104.12.0.m.1	$104$	$2$	$2$	$0$
104.12.0.p.1	$104$	$2$	$2$	$0$
116.12.0.e.1	$116$	$2$	$2$	$0$
116.12.0.f.1	$116$	$2$	$2$	$0$
116.180.13.b.1	$116$	$30$	$30$	$13$
120.12.0.m.1	$120$	$2$	$2$	$0$
120.12.0.p.1	$120$	$2$	$2$	$0$
124.12.0.e.1	$124$	$2$	$2$	$0$
124.12.0.f.1	$124$	$2$	$2$	$0$
124.192.14.b.1	$124$	$32$	$32$	$14$
132.12.0.e.1	$132$	$2$	$2$	$0$
132.12.0.f.1	$132$	$2$	$2$	$0$
136.12.0.m.1	$136$	$2$	$2$	$0$
136.12.0.p.1	$136$	$2$	$2$	$0$
140.12.0.e.1	$140$	$2$	$2$	$0$
140.12.0.f.1	$140$	$2$	$2$	$0$
148.12.0.e.1	$148$	$2$	$2$	$0$
148.12.0.f.1	$148$	$2$	$2$	$0$
148.228.17.b.1	$148$	$38$	$38$	$17$
152.12.0.m.1	$152$	$2$	$2$	$0$
152.12.0.p.1	$152$	$2$	$2$	$0$
156.12.0.e.1	$156$	$2$	$2$	$0$
156.12.0.f.1	$156$	$2$	$2$	$0$
164.12.0.e.1	$164$	$2$	$2$	$0$
164.12.0.f.1	$164$	$2$	$2$	$0$
164.252.19.b.1	$164$	$42$	$42$	$19$
168.12.0.m.1	$168$	$2$	$2$	$0$
168.12.0.p.1	$168$	$2$	$2$	$0$
172.12.0.e.1	$172$	$2$	$2$	$0$
172.12.0.f.1	$172$	$2$	$2$	$0$
172.264.20.b.1	$172$	$44$	$44$	$20$
184.12.0.m.1	$184$	$2$	$2$	$0$
184.12.0.p.1	$184$	$2$	$2$	$0$
188.12.0.e.1	$188$	$2$	$2$	$0$
188.12.0.f.1	$188$	$2$	$2$	$0$
188.288.22.b.1	$188$	$48$	$48$	$22$
204.12.0.e.1	$204$	$2$	$2$	$0$
204.12.0.f.1	$204$	$2$	$2$	$0$
212.12.0.e.1	$212$	$2$	$2$	$0$
212.12.0.f.1	$212$	$2$	$2$	$0$
220.12.0.e.1	$220$	$2$	$2$	$0$
220.12.0.f.1	$220$	$2$	$2$	$0$
228.12.0.e.1	$228$	$2$	$2$	$0$
228.12.0.f.1	$228$	$2$	$2$	$0$
232.12.0.m.1	$232$	$2$	$2$	$0$
232.12.0.p.1	$232$	$2$	$2$	$0$
236.12.0.e.1	$236$	$2$	$2$	$0$
236.12.0.f.1	$236$	$2$	$2$	$0$
244.12.0.e.1	$244$	$2$	$2$	$0$
244.12.0.f.1	$244$	$2$	$2$	$0$
248.12.0.m.1	$248$	$2$	$2$	$0$
248.12.0.p.1	$248$	$2$	$2$	$0$
260.12.0.e.1	$260$	$2$	$2$	$0$
260.12.0.f.1	$260$	$2$	$2$	$0$
264.12.0.m.1	$264$	$2$	$2$	$0$
264.12.0.p.1	$264$	$2$	$2$	$0$
268.12.0.e.1	$268$	$2$	$2$	$0$
268.12.0.f.1	$268$	$2$	$2$	$0$
276.12.0.e.1	$276$	$2$	$2$	$0$
276.12.0.f.1	$276$	$2$	$2$	$0$
280.12.0.m.1	$280$	$2$	$2$	$0$
280.12.0.p.1	$280$	$2$	$2$	$0$
284.12.0.e.1	$284$	$2$	$2$	$0$
284.12.0.f.1	$284$	$2$	$2$	$0$
292.12.0.e.1	$292$	$2$	$2$	$0$
292.12.0.f.1	$292$	$2$	$2$	$0$
296.12.0.m.1	$296$	$2$	$2$	$0$
296.12.0.p.1	$296$	$2$	$2$	$0$
308.12.0.e.1	$308$	$2$	$2$	$0$
308.12.0.f.1	$308$	$2$	$2$	$0$
312.12.0.m.1	$312$	$2$	$2$	$0$
312.12.0.p.1	$312$	$2$	$2$	$0$
316.12.0.e.1	$316$	$2$	$2$	$0$
316.12.0.f.1	$316$	$2$	$2$	$0$
328.12.0.m.1	$328$	$2$	$2$	$0$
328.12.0.p.1	$328$	$2$	$2$	$0$
332.12.0.e.1	$332$	$2$	$2$	$0$
332.12.0.f.1	$332$	$2$	$2$	$0$