Modular curve 6.36.0.a.1

• Label: 6.36.0.a.1
• Level: $6$
• Index: $36$
• Genus: $0$
• Analytic rank: $0$
• Cusps: $8$
• $\Q$-cusps: $4$

Invariants

Level:	$6$		$\SL_2$-level:	$6$
Index:	$36$		$\PSL_2$-index:	$36$
Genus:	$0 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (of which $4$ are rational)		Cusp widths	$3^{4}\cdot6^{4}$		Cusp orbits	$1^{4}\cdot2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
$\Q$-gonality:	$1$
$\overline{\Q}$-gonality:	$1$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	6K0
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	6.36.0.1

Level structure

$\GL_2(\Z/6\Z)$-generators:	$\begin{bmatrix}1&3\\0&1\end{bmatrix}$, $\begin{bmatrix}5&0\\0&1\end{bmatrix}$, $\begin{bmatrix}5&0\\0&5\end{bmatrix}$
$\GL_2(\Z/6\Z)$-subgroup:	$C_2^3$
Contains $-I$:	yes
Quadratic refinements:	6.72.0-6.a.1.1, 6.72.0-6.a.1.2, 12.72.0-6.a.1.1, 12.72.0-6.a.1.2, 12.72.0-6.a.1.3, 12.72.0-6.a.1.4, 12.72.0-6.a.1.5, 12.72.0-6.a.1.6, 24.72.0-6.a.1.1, 24.72.0-6.a.1.2, 24.72.0-6.a.1.3, 24.72.0-6.a.1.4, 24.72.0-6.a.1.5, 24.72.0-6.a.1.6, 24.72.0-6.a.1.7, 24.72.0-6.a.1.8, 30.72.0-6.a.1.1, 30.72.0-6.a.1.2, 42.72.0-6.a.1.1, 42.72.0-6.a.1.2, 60.72.0-6.a.1.1, 60.72.0-6.a.1.2, 60.72.0-6.a.1.3, 60.72.0-6.a.1.4, 60.72.0-6.a.1.5, 60.72.0-6.a.1.6, 66.72.0-6.a.1.1, 66.72.0-6.a.1.2, 78.72.0-6.a.1.1, 78.72.0-6.a.1.2, 84.72.0-6.a.1.1, 84.72.0-6.a.1.2, 84.72.0-6.a.1.3, 84.72.0-6.a.1.4, 84.72.0-6.a.1.5, 84.72.0-6.a.1.6, 102.72.0-6.a.1.1, 102.72.0-6.a.1.2, 114.72.0-6.a.1.1, 114.72.0-6.a.1.2, 120.72.0-6.a.1.1, 120.72.0-6.a.1.2, 120.72.0-6.a.1.3, 120.72.0-6.a.1.4, 120.72.0-6.a.1.5, 120.72.0-6.a.1.6, 120.72.0-6.a.1.7, 120.72.0-6.a.1.8, 132.72.0-6.a.1.1, 132.72.0-6.a.1.2, 132.72.0-6.a.1.3, 132.72.0-6.a.1.4, 132.72.0-6.a.1.5, 132.72.0-6.a.1.6, 138.72.0-6.a.1.1, 138.72.0-6.a.1.2, 156.72.0-6.a.1.1, 156.72.0-6.a.1.2, 156.72.0-6.a.1.3, 156.72.0-6.a.1.4, 156.72.0-6.a.1.5, 156.72.0-6.a.1.6, 168.72.0-6.a.1.1, 168.72.0-6.a.1.2, 168.72.0-6.a.1.3, 168.72.0-6.a.1.4, 168.72.0-6.a.1.5, 168.72.0-6.a.1.6, 168.72.0-6.a.1.7, 168.72.0-6.a.1.8, 174.72.0-6.a.1.1, 174.72.0-6.a.1.2, 186.72.0-6.a.1.1, 186.72.0-6.a.1.2, 204.72.0-6.a.1.1, 204.72.0-6.a.1.2, 204.72.0-6.a.1.3, 204.72.0-6.a.1.4, 204.72.0-6.a.1.5, 204.72.0-6.a.1.6, 210.72.0-6.a.1.1, 210.72.0-6.a.1.2, 222.72.0-6.a.1.1, 222.72.0-6.a.1.2, 228.72.0-6.a.1.1, 228.72.0-6.a.1.2, 228.72.0-6.a.1.3, 228.72.0-6.a.1.4, 228.72.0-6.a.1.5, 228.72.0-6.a.1.6, 246.72.0-6.a.1.1, 246.72.0-6.a.1.2, 258.72.0-6.a.1.1, 258.72.0-6.a.1.2, 264.72.0-6.a.1.1, 264.72.0-6.a.1.2, 264.72.0-6.a.1.3, 264.72.0-6.a.1.4, 264.72.0-6.a.1.5, 264.72.0-6.a.1.6, 264.72.0-6.a.1.7, 264.72.0-6.a.1.8, 276.72.0-6.a.1.1, 276.72.0-6.a.1.2, 276.72.0-6.a.1.3, 276.72.0-6.a.1.4, 276.72.0-6.a.1.5, 276.72.0-6.a.1.6, 282.72.0-6.a.1.1, 282.72.0-6.a.1.2, 312.72.0-6.a.1.1, 312.72.0-6.a.1.2, 312.72.0-6.a.1.3, 312.72.0-6.a.1.4, 312.72.0-6.a.1.5, 312.72.0-6.a.1.6, 312.72.0-6.a.1.7, 312.72.0-6.a.1.8, 318.72.0-6.a.1.1, 318.72.0-6.a.1.2, 330.72.0-6.a.1.1, 330.72.0-6.a.1.2
Cyclic 6-isogeny field degree:	$1$
Cyclic 6-torsion field degree:	$2$
Full 6-torsion field degree:	$8$

Models

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

Rational points

This modular curve has infinitely many rational points, including 46 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 36 to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{x^{36}(x^{3}-2y^{3})^{3}(x^{3}+6xy^{2}-2y^{3})^{3}(x^{6}-6x^{4}y^{2}-4x^{3}y^{3}+36x^{2}y^{4}+12xy^{5}+4y^{6})^{3}}{y^{6}x^{39}(x-2y)^{3}(x+y)^{6}(x^{2}-xy+y^{2})^{6}(x^{2}+2xy+4y^{2})^{3}}$

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
$X_0(2)$	$2$	$12$	$12$	$0$	$0$
$X_{\mathrm{sp}}(3)$	$3$	$3$	$3$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
$X_{\mathrm{sp}}(3)$	$3$	$3$	$3$	$0$	$0$
$X_0(6)$	$6$	$3$	$3$	$0$	$0$
6.18.0.b.1	$6$	$2$	$2$	$0$	$0$

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

Covering curve	Level	Index	Degree	Genus
$X_{\mathrm{sp}}(6)$	$6$	$2$	$2$	$1$
6.72.1.b.1	$6$	$2$	$2$	$1$
12.72.1.b.1	$12$	$2$	$2$	$1$
12.72.1.d.1	$12$	$2$	$2$	$1$
12.72.1.f.1	$12$	$2$	$2$	$1$
12.72.1.h.1	$12$	$2$	$2$	$1$
12.72.1.i.1	$12$	$2$	$2$	$1$
12.72.1.l.1	$12$	$2$	$2$	$1$
12.72.3.ce.1	$12$	$2$	$2$	$3$
12.72.3.cf.1	$12$	$2$	$2$	$3$
12.72.3.cy.1	$12$	$2$	$2$	$3$
12.72.3.da.1	$12$	$2$	$2$	$3$
18.108.2.a.1	$18$	$3$	$3$	$2$
18.108.2.b.1	$18$	$3$	$3$	$2$
18.108.2.c.1	$18$	$3$	$3$	$2$
18.108.4.c.1	$18$	$3$	$3$	$4$
18.108.6.a.1	$18$	$3$	$3$	$6$
24.72.1.c.1	$24$	$2$	$2$	$1$
24.72.1.h.1	$24$	$2$	$2$	$1$
24.72.1.n.1	$24$	$2$	$2$	$1$
24.72.1.t.1	$24$	$2$	$2$	$1$
24.72.1.y.1	$24$	$2$	$2$	$1$
24.72.1.bb.1	$24$	$2$	$2$	$1$
24.72.1.bk.1	$24$	$2$	$2$	$1$
24.72.1.bn.1	$24$	$2$	$2$	$1$
24.72.3.qm.1	$24$	$2$	$2$	$3$
24.72.3.qp.1	$24$	$2$	$2$	$3$
24.72.3.ua.1	$24$	$2$	$2$	$3$
24.72.3.ug.1	$24$	$2$	$2$	$3$
30.72.1.c.1	$30$	$2$	$2$	$1$
30.72.1.e.1	$30$	$2$	$2$	$1$
30.180.12.a.1	$30$	$5$	$5$	$12$
30.216.11.a.1	$30$	$6$	$6$	$11$
30.360.23.a.1	$30$	$10$	$10$	$23$
42.72.1.d.1	$42$	$2$	$2$	$1$
42.72.1.e.1	$42$	$2$	$2$	$1$
42.288.17.a.1	$42$	$8$	$8$	$17$
42.756.52.a.1	$42$	$21$	$21$	$52$
42.1008.69.a.1	$42$	$28$	$28$	$69$
60.72.1.o.1	$60$	$2$	$2$	$1$
60.72.1.r.1	$60$	$2$	$2$	$1$
60.72.1.s.1	$60$	$2$	$2$	$1$
60.72.1.bc.1	$60$	$2$	$2$	$1$
60.72.1.bd.1	$60$	$2$	$2$	$1$
60.72.1.bg.1	$60$	$2$	$2$	$1$
60.72.3.lc.1	$60$	$2$	$2$	$3$
60.72.3.ld.1	$60$	$2$	$2$	$3$
60.72.3.np.1	$60$	$2$	$2$	$3$
60.72.3.nq.1	$60$	$2$	$2$	$3$
66.72.1.b.1	$66$	$2$	$2$	$1$
66.72.1.c.1	$66$	$2$	$2$	$1$
66.432.29.a.1	$66$	$12$	$12$	$29$
66.1980.146.a.1	$66$	$55$	$55$	$146$
66.1980.146.b.1	$66$	$55$	$55$	$146$
66.2376.175.a.1	$66$	$66$	$66$	$175$
78.72.1.d.1	$78$	$2$	$2$	$1$
78.72.1.e.1	$78$	$2$	$2$	$1$
84.72.1.j.1	$84$	$2$	$2$	$1$
84.72.1.m.1	$84$	$2$	$2$	$1$
84.72.1.n.1	$84$	$2$	$2$	$1$
84.72.1.q.1	$84$	$2$	$2$	$1$
84.72.1.r.1	$84$	$2$	$2$	$1$
84.72.1.u.1	$84$	$2$	$2$	$1$
84.72.3.jw.1	$84$	$2$	$2$	$3$
84.72.3.jx.1	$84$	$2$	$2$	$3$
84.72.3.ma.1	$84$	$2$	$2$	$3$
84.72.3.mb.1	$84$	$2$	$2$	$3$
102.72.1.b.1	$102$	$2$	$2$	$1$
102.72.1.c.1	$102$	$2$	$2$	$1$
114.72.1.d.1	$114$	$2$	$2$	$1$
114.72.1.e.1	$114$	$2$	$2$	$1$
120.72.1.bw.1	$120$	$2$	$2$	$1$
120.72.1.bz.1	$120$	$2$	$2$	$1$
120.72.1.ci.1	$120$	$2$	$2$	$1$
120.72.1.cl.1	$120$	$2$	$2$	$1$
120.72.1.ds.1	$120$	$2$	$2$	$1$
120.72.1.dv.1	$120$	$2$	$2$	$1$
120.72.1.ee.1	$120$	$2$	$2$	$1$
120.72.1.eh.1	$120$	$2$	$2$	$1$
120.72.3.dae.1	$120$	$2$	$2$	$3$
120.72.3.dah.1	$120$	$2$	$2$	$3$
120.72.3.dnu.1	$120$	$2$	$2$	$3$
120.72.3.dnx.1	$120$	$2$	$2$	$3$
126.108.2.a.1	$126$	$3$	$3$	$2$
126.108.2.b.1	$126$	$3$	$3$	$2$
126.108.2.c.1	$126$	$3$	$3$	$2$
132.72.1.h.1	$132$	$2$	$2$	$1$
132.72.1.k.1	$132$	$2$	$2$	$1$
132.72.1.l.1	$132$	$2$	$2$	$1$
132.72.1.o.1	$132$	$2$	$2$	$1$
132.72.1.p.1	$132$	$2$	$2$	$1$
132.72.1.s.1	$132$	$2$	$2$	$1$
132.72.3.jw.1	$132$	$2$	$2$	$3$
132.72.3.jx.1	$132$	$2$	$2$	$3$
132.72.3.ma.1	$132$	$2$	$2$	$3$
132.72.3.mb.1	$132$	$2$	$2$	$3$
138.72.1.b.1	$138$	$2$	$2$	$1$
138.72.1.c.1	$138$	$2$	$2$	$1$
156.72.1.j.1	$156$	$2$	$2$	$1$
156.72.1.m.1	$156$	$2$	$2$	$1$
156.72.1.n.1	$156$	$2$	$2$	$1$
156.72.1.q.1	$156$	$2$	$2$	$1$
156.72.1.r.1	$156$	$2$	$2$	$1$
156.72.1.u.1	$156$	$2$	$2$	$1$
156.72.3.jw.1	$156$	$2$	$2$	$3$
156.72.3.jx.1	$156$	$2$	$2$	$3$
156.72.3.ma.1	$156$	$2$	$2$	$3$
156.72.3.mb.1	$156$	$2$	$2$	$3$
168.72.1.bc.1	$168$	$2$	$2$	$1$
168.72.1.bf.1	$168$	$2$	$2$	$1$
168.72.1.bo.1	$168$	$2$	$2$	$1$
168.72.1.br.1	$168$	$2$	$2$	$1$
168.72.1.ca.1	$168$	$2$	$2$	$1$
168.72.1.cd.1	$168$	$2$	$2$	$1$
168.72.1.cm.1	$168$	$2$	$2$	$1$
168.72.1.cp.1	$168$	$2$	$2$	$1$
168.72.3.cwa.1	$168$	$2$	$2$	$3$
168.72.3.cwd.1	$168$	$2$	$2$	$3$
168.72.3.diq.1	$168$	$2$	$2$	$3$
168.72.3.dit.1	$168$	$2$	$2$	$3$
174.72.1.b.1	$174$	$2$	$2$	$1$
174.72.1.c.1	$174$	$2$	$2$	$1$
186.72.1.d.1	$186$	$2$	$2$	$1$
186.72.1.e.1	$186$	$2$	$2$	$1$
204.72.1.h.1	$204$	$2$	$2$	$1$
204.72.1.k.1	$204$	$2$	$2$	$1$
204.72.1.l.1	$204$	$2$	$2$	$1$
204.72.1.o.1	$204$	$2$	$2$	$1$
204.72.1.p.1	$204$	$2$	$2$	$1$
204.72.1.s.1	$204$	$2$	$2$	$1$
204.72.3.jw.1	$204$	$2$	$2$	$3$
204.72.3.jx.1	$204$	$2$	$2$	$3$
204.72.3.ma.1	$204$	$2$	$2$	$3$
204.72.3.mb.1	$204$	$2$	$2$	$3$
210.72.1.g.1	$210$	$2$	$2$	$1$
210.72.1.i.1	$210$	$2$	$2$	$1$
222.72.1.d.1	$222$	$2$	$2$	$1$
222.72.1.e.1	$222$	$2$	$2$	$1$
228.72.1.j.1	$228$	$2$	$2$	$1$
228.72.1.m.1	$228$	$2$	$2$	$1$
228.72.1.n.1	$228$	$2$	$2$	$1$
228.72.1.q.1	$228$	$2$	$2$	$1$
228.72.1.r.1	$228$	$2$	$2$	$1$
228.72.1.u.1	$228$	$2$	$2$	$1$
228.72.3.jw.1	$228$	$2$	$2$	$3$
228.72.3.jx.1	$228$	$2$	$2$	$3$
228.72.3.ma.1	$228$	$2$	$2$	$3$
228.72.3.mb.1	$228$	$2$	$2$	$3$
234.108.2.a.1	$234$	$3$	$3$	$2$
234.108.2.b.1	$234$	$3$	$3$	$2$
234.108.2.c.1	$234$	$3$	$3$	$2$
246.72.1.b.1	$246$	$2$	$2$	$1$
246.72.1.c.1	$246$	$2$	$2$	$1$
258.72.1.d.1	$258$	$2$	$2$	$1$
258.72.1.e.1	$258$	$2$	$2$	$1$
264.72.1.y.1	$264$	$2$	$2$	$1$
264.72.1.bb.1	$264$	$2$	$2$	$1$
264.72.1.bk.1	$264$	$2$	$2$	$1$
264.72.1.bn.1	$264$	$2$	$2$	$1$
264.72.1.bw.1	$264$	$2$	$2$	$1$
264.72.1.bz.1	$264$	$2$	$2$	$1$
264.72.1.ci.1	$264$	$2$	$2$	$1$
264.72.1.cl.1	$264$	$2$	$2$	$1$
264.72.3.cwa.1	$264$	$2$	$2$	$3$
264.72.3.cwd.1	$264$	$2$	$2$	$3$
264.72.3.diq.1	$264$	$2$	$2$	$3$
264.72.3.dit.1	$264$	$2$	$2$	$3$
276.72.1.h.1	$276$	$2$	$2$	$1$
276.72.1.k.1	$276$	$2$	$2$	$1$
276.72.1.l.1	$276$	$2$	$2$	$1$
276.72.1.o.1	$276$	$2$	$2$	$1$
276.72.1.p.1	$276$	$2$	$2$	$1$
276.72.1.s.1	$276$	$2$	$2$	$1$
276.72.3.jw.1	$276$	$2$	$2$	$3$
276.72.3.jx.1	$276$	$2$	$2$	$3$
276.72.3.ma.1	$276$	$2$	$2$	$3$
276.72.3.mb.1	$276$	$2$	$2$	$3$
282.72.1.b.1	$282$	$2$	$2$	$1$
282.72.1.c.1	$282$	$2$	$2$	$1$
312.72.1.bc.1	$312$	$2$	$2$	$1$
312.72.1.bf.1	$312$	$2$	$2$	$1$
312.72.1.bo.1	$312$	$2$	$2$	$1$
312.72.1.br.1	$312$	$2$	$2$	$1$
312.72.1.ca.1	$312$	$2$	$2$	$1$
312.72.1.cd.1	$312$	$2$	$2$	$1$
312.72.1.cm.1	$312$	$2$	$2$	$1$
312.72.1.cp.1	$312$	$2$	$2$	$1$
312.72.3.cwa.1	$312$	$2$	$2$	$3$
312.72.3.cwd.1	$312$	$2$	$2$	$3$
312.72.3.diq.1	$312$	$2$	$2$	$3$
312.72.3.dit.1	$312$	$2$	$2$	$3$
318.72.1.b.1	$318$	$2$	$2$	$1$
318.72.1.c.1	$318$	$2$	$2$	$1$
330.72.1.c.1	$330$	$2$	$2$	$1$
330.72.1.e.1	$330$	$2$	$2$	$1$