Modular curve $X_{\mathrm{sp}}^+(6)$

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

Invariants

Level:	$6$		$\SL_2$-level:	$6$
Index:	$36$		$\PSL_2$-index:	$36$
Genus:	$0 = 1 + \frac{ 36 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps:	$6$ (of which $4$ are rational)		Cusp widths	$6^{6}$		Cusp orbits	$1^{4}\cdot2$
Elliptic points:	$4$ 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:	6L0
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	6.36.0.2

Level structure

$\GL_2(\Z/6\Z)$-generators:	$\begin{bmatrix}0&5\\1&0\end{bmatrix}$, $\begin{bmatrix}1&0\\0&5\end{bmatrix}$
$\GL_2(\Z/6\Z)$-subgroup:	$D_4$
Contains $-I$:	yes
Quadratic refinements:	none in database
Cyclic 6-isogeny field degree:	$2$
Cyclic 6-torsion field degree:	$4$
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 6 stored non-cuspidal points.

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle \frac{x^{36}(x^{3}-6xy^{2}-6y^{3})^{3}(x^{3}+6x^{2}y+12xy^{2}+6y^{3})^{3}(x^{6}+6x^{5}y+18x^{4}y^{2}+36x^{3}y^{3}+48x^{2}y^{4}+36xy^{5}+12y^{6})^{3}}{y^{6}x^{42}(x+y)^{6}(x+2y)^{6}(x^{2}+3xy+3y^{2})^{6}}$

Modular covers

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
6.12.0.b.1	$6$	$3$	$3$	$0$	$0$
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.c.1	$12$	$2$	$2$	$1$
12.72.1.n.1	$12$	$2$	$2$	$1$
12.72.1.q.1	$12$	$2$	$2$	$1$
12.72.1.r.1	$12$	$2$	$2$	$1$
12.72.1.t.1	$12$	$2$	$2$	$1$
12.72.1.v.1	$12$	$2$	$2$	$1$
12.72.3.cb.1	$12$	$2$	$2$	$3$
12.72.3.cd.1	$12$	$2$	$2$	$3$
12.72.3.cw.1	$12$	$2$	$2$	$3$
12.72.3.cx.1	$12$	$2$	$2$	$3$
18.108.3.a.1	$18$	$3$	$3$	$3$
18.108.3.b.1	$18$	$3$	$3$	$3$
18.108.4.i.1	$18$	$3$	$3$	$4$
24.72.1.f.1	$24$	$2$	$2$	$1$
24.72.1.l.1	$24$	$2$	$2$	$1$
24.72.1.bs.1	$24$	$2$	$2$	$1$
24.72.1.bv.1	$24$	$2$	$2$	$1$
24.72.1.ce.1	$24$	$2$	$2$	$1$
24.72.1.ch.1	$24$	$2$	$2$	$1$
24.72.1.cm.1	$24$	$2$	$2$	$1$
24.72.1.cs.1	$24$	$2$	$2$	$1$
24.72.3.qf.1	$24$	$2$	$2$	$3$
24.72.3.ql.1	$24$	$2$	$2$	$3$
24.72.3.tw.1	$24$	$2$	$2$	$3$
24.72.3.tz.1	$24$	$2$	$2$	$3$
30.72.1.k.1	$30$	$2$	$2$	$1$
30.72.1.l.1	$30$	$2$	$2$	$1$
30.180.12.s.1	$30$	$5$	$5$	$12$
30.216.11.t.1	$30$	$6$	$6$	$11$
30.360.23.cl.1	$30$	$10$	$10$	$23$
42.72.1.g.1	$42$	$2$	$2$	$1$
42.72.1.h.1	$42$	$2$	$2$	$1$
42.288.19.cn.1	$42$	$8$	$8$	$19$
42.756.50.b.1	$42$	$21$	$21$	$50$
42.1008.69.bz.1	$42$	$28$	$28$	$69$
60.72.1.cy.1	$60$	$2$	$2$	$1$
60.72.1.df.1	$60$	$2$	$2$	$1$
60.72.1.ep.1	$60$	$2$	$2$	$1$
60.72.1.eq.1	$60$	$2$	$2$	$1$
60.72.1.ex.1	$60$	$2$	$2$	$1$
60.72.1.ey.1	$60$	$2$	$2$	$1$
60.72.3.ue.1	$60$	$2$	$2$	$3$
60.72.3.uf.1	$60$	$2$	$2$	$3$
60.72.3.uu.1	$60$	$2$	$2$	$3$
60.72.3.uv.1	$60$	$2$	$2$	$3$
66.72.1.e.1	$66$	$2$	$2$	$1$
66.72.1.f.1	$66$	$2$	$2$	$1$
66.432.31.r.1	$66$	$12$	$12$	$31$
66.1980.144.d.1	$66$	$55$	$55$	$144$
66.1980.148.i.1	$66$	$55$	$55$	$148$
66.2376.175.bj.1	$66$	$66$	$66$	$175$
78.72.1.g.1	$78$	$2$	$2$	$1$
78.72.1.h.1	$78$	$2$	$2$	$1$
84.72.1.bk.1	$84$	$2$	$2$	$1$
84.72.1.br.1	$84$	$2$	$2$	$1$
84.72.1.cc.1	$84$	$2$	$2$	$1$
84.72.1.cd.1	$84$	$2$	$2$	$1$
84.72.1.ck.1	$84$	$2$	$2$	$1$
84.72.1.cl.1	$84$	$2$	$2$	$1$
84.72.3.qk.1	$84$	$2$	$2$	$3$
84.72.3.ql.1	$84$	$2$	$2$	$3$
84.72.3.ra.1	$84$	$2$	$2$	$3$
84.72.3.rb.1	$84$	$2$	$2$	$3$
102.72.1.e.1	$102$	$2$	$2$	$1$
102.72.1.f.1	$102$	$2$	$2$	$1$
114.72.1.g.1	$114$	$2$	$2$	$1$
114.72.1.h.1	$114$	$2$	$2$	$1$
120.72.1.ka.1	$120$	$2$	$2$	$1$
120.72.1.kd.1	$120$	$2$	$2$	$1$
120.72.1.ky.1	$120$	$2$	$2$	$1$
120.72.1.lb.1	$120$	$2$	$2$	$1$
120.72.1.pe.1	$120$	$2$	$2$	$1$
120.72.1.ph.1	$120$	$2$	$2$	$1$
120.72.1.qc.1	$120$	$2$	$2$	$1$
120.72.1.qf.1	$120$	$2$	$2$	$1$
120.72.3.evc.1	$120$	$2$	$2$	$3$
120.72.3.evf.1	$120$	$2$	$2$	$3$
120.72.3.eye.1	$120$	$2$	$2$	$3$
120.72.3.eyh.1	$120$	$2$	$2$	$3$
132.72.1.bi.1	$132$	$2$	$2$	$1$
132.72.1.bp.1	$132$	$2$	$2$	$1$
132.72.1.ca.1	$132$	$2$	$2$	$1$
132.72.1.cb.1	$132$	$2$	$2$	$1$
132.72.1.ci.1	$132$	$2$	$2$	$1$
132.72.1.cj.1	$132$	$2$	$2$	$1$
132.72.3.qk.1	$132$	$2$	$2$	$3$
132.72.3.ql.1	$132$	$2$	$2$	$3$
132.72.3.ra.1	$132$	$2$	$2$	$3$
132.72.3.rb.1	$132$	$2$	$2$	$3$
138.72.1.e.1	$138$	$2$	$2$	$1$
138.72.1.f.1	$138$	$2$	$2$	$1$
156.72.1.bk.1	$156$	$2$	$2$	$1$
156.72.1.br.1	$156$	$2$	$2$	$1$
156.72.1.cc.1	$156$	$2$	$2$	$1$
156.72.1.cd.1	$156$	$2$	$2$	$1$
156.72.1.ck.1	$156$	$2$	$2$	$1$
156.72.1.cl.1	$156$	$2$	$2$	$1$
156.72.3.qk.1	$156$	$2$	$2$	$3$
156.72.3.ql.1	$156$	$2$	$2$	$3$
156.72.3.ra.1	$156$	$2$	$2$	$3$
156.72.3.rb.1	$156$	$2$	$2$	$3$
168.72.1.eq.1	$168$	$2$	$2$	$1$
168.72.1.et.1	$168$	$2$	$2$	$1$
168.72.1.fo.1	$168$	$2$	$2$	$1$
168.72.1.fr.1	$168$	$2$	$2$	$1$
168.72.1.gy.1	$168$	$2$	$2$	$1$
168.72.1.hb.1	$168$	$2$	$2$	$1$
168.72.1.hw.1	$168$	$2$	$2$	$1$
168.72.1.hz.1	$168$	$2$	$2$	$1$
168.72.3.ejk.1	$168$	$2$	$2$	$3$
168.72.3.ejn.1	$168$	$2$	$2$	$3$
168.72.3.emm.1	$168$	$2$	$2$	$3$
168.72.3.emp.1	$168$	$2$	$2$	$3$
174.72.1.e.1	$174$	$2$	$2$	$1$
174.72.1.f.1	$174$	$2$	$2$	$1$
186.72.1.g.1	$186$	$2$	$2$	$1$
186.72.1.h.1	$186$	$2$	$2$	$1$
204.72.1.bi.1	$204$	$2$	$2$	$1$
204.72.1.bp.1	$204$	$2$	$2$	$1$
204.72.1.cc.1	$204$	$2$	$2$	$1$
204.72.1.cd.1	$204$	$2$	$2$	$1$
204.72.1.ck.1	$204$	$2$	$2$	$1$
204.72.1.cl.1	$204$	$2$	$2$	$1$
204.72.3.qk.1	$204$	$2$	$2$	$3$
204.72.3.ql.1	$204$	$2$	$2$	$3$
204.72.3.ra.1	$204$	$2$	$2$	$3$
204.72.3.rb.1	$204$	$2$	$2$	$3$
210.72.1.bm.1	$210$	$2$	$2$	$1$
210.72.1.bn.1	$210$	$2$	$2$	$1$
222.72.1.g.1	$222$	$2$	$2$	$1$
222.72.1.h.1	$222$	$2$	$2$	$1$
228.72.1.bk.1	$228$	$2$	$2$	$1$
228.72.1.br.1	$228$	$2$	$2$	$1$
228.72.1.cc.1	$228$	$2$	$2$	$1$
228.72.1.cd.1	$228$	$2$	$2$	$1$
228.72.1.ck.1	$228$	$2$	$2$	$1$
228.72.1.cl.1	$228$	$2$	$2$	$1$
228.72.3.qk.1	$228$	$2$	$2$	$3$
228.72.3.ql.1	$228$	$2$	$2$	$3$
228.72.3.ra.1	$228$	$2$	$2$	$3$
228.72.3.rb.1	$228$	$2$	$2$	$3$
246.72.1.e.1	$246$	$2$	$2$	$1$
246.72.1.f.1	$246$	$2$	$2$	$1$
258.72.1.g.1	$258$	$2$	$2$	$1$
258.72.1.h.1	$258$	$2$	$2$	$1$
264.72.1.em.1	$264$	$2$	$2$	$1$
264.72.1.ep.1	$264$	$2$	$2$	$1$
264.72.1.fk.1	$264$	$2$	$2$	$1$
264.72.1.fn.1	$264$	$2$	$2$	$1$
264.72.1.gu.1	$264$	$2$	$2$	$1$
264.72.1.gx.1	$264$	$2$	$2$	$1$
264.72.1.hs.1	$264$	$2$	$2$	$1$
264.72.1.hv.1	$264$	$2$	$2$	$1$
264.72.3.ejk.1	$264$	$2$	$2$	$3$
264.72.3.ejn.1	$264$	$2$	$2$	$3$
264.72.3.emm.1	$264$	$2$	$2$	$3$
264.72.3.emp.1	$264$	$2$	$2$	$3$
276.72.1.bi.1	$276$	$2$	$2$	$1$
276.72.1.bp.1	$276$	$2$	$2$	$1$
276.72.1.ca.1	$276$	$2$	$2$	$1$
276.72.1.cb.1	$276$	$2$	$2$	$1$
276.72.1.ci.1	$276$	$2$	$2$	$1$
276.72.1.cj.1	$276$	$2$	$2$	$1$
276.72.3.qk.1	$276$	$2$	$2$	$3$
276.72.3.ql.1	$276$	$2$	$2$	$3$
276.72.3.ra.1	$276$	$2$	$2$	$3$
276.72.3.rb.1	$276$	$2$	$2$	$3$
282.72.1.e.1	$282$	$2$	$2$	$1$
282.72.1.f.1	$282$	$2$	$2$	$1$
312.72.1.eq.1	$312$	$2$	$2$	$1$
312.72.1.et.1	$312$	$2$	$2$	$1$
312.72.1.fo.1	$312$	$2$	$2$	$1$
312.72.1.fr.1	$312$	$2$	$2$	$1$
312.72.1.gy.1	$312$	$2$	$2$	$1$
312.72.1.hb.1	$312$	$2$	$2$	$1$
312.72.1.hw.1	$312$	$2$	$2$	$1$
312.72.1.hz.1	$312$	$2$	$2$	$1$
312.72.3.ejk.1	$312$	$2$	$2$	$3$
312.72.3.ejn.1	$312$	$2$	$2$	$3$
312.72.3.emm.1	$312$	$2$	$2$	$3$
312.72.3.emp.1	$312$	$2$	$2$	$3$
318.72.1.e.1	$318$	$2$	$2$	$1$
318.72.1.f.1	$318$	$2$	$2$	$1$
330.72.1.bi.1	$330$	$2$	$2$	$1$
330.72.1.bj.1	$330$	$2$	$2$	$1$