Modular curve 10.36.1.a.1

• Label: 10.36.1.a.1
• Level: $10$
• Index: $36$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $6$
• $\Q$-cusps: $6$

Invariants

Level:	$10$		$\SL_2$-level:	$10$		Newform level:	$20$
Index:	$36$		$\PSL_2$-index:	$36$
Genus:	$1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps:	$6$ (all of which are rational)		Cusp widths	$2^{3}\cdot10^{3}$		Cusp orbits	$1^{6}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$6$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	10G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	10.36.1.1

Level structure

$\GL_2(\Z/10\Z)$-generators:	$\begin{bmatrix}3&0\\0&7\end{bmatrix}$, $\begin{bmatrix}3&8\\0&9\end{bmatrix}$
$\GL_2(\Z/10\Z)$-subgroup:	$C_4\times F_5$
Contains $-I$:	yes
Quadratic refinements:	20.72.1-10.a.1.1, 20.72.1-10.a.1.2, 20.72.1-10.a.1.3, 20.72.1-10.a.1.4, 20.72.1-10.a.1.5, 20.72.1-10.a.1.6, 20.72.1-10.a.1.7, 20.72.1-10.a.1.8, 40.72.1-10.a.1.1, 40.72.1-10.a.1.2, 40.72.1-10.a.1.3, 40.72.1-10.a.1.4, 40.72.1-10.a.1.5, 40.72.1-10.a.1.6, 40.72.1-10.a.1.7, 40.72.1-10.a.1.8, 60.72.1-10.a.1.1, 60.72.1-10.a.1.2, 60.72.1-10.a.1.3, 60.72.1-10.a.1.4, 60.72.1-10.a.1.5, 60.72.1-10.a.1.6, 60.72.1-10.a.1.7, 60.72.1-10.a.1.8, 120.72.1-10.a.1.1, 120.72.1-10.a.1.2, 120.72.1-10.a.1.3, 120.72.1-10.a.1.4, 120.72.1-10.a.1.5, 120.72.1-10.a.1.6, 120.72.1-10.a.1.7, 120.72.1-10.a.1.8, 140.72.1-10.a.1.1, 140.72.1-10.a.1.2, 140.72.1-10.a.1.3, 140.72.1-10.a.1.4, 140.72.1-10.a.1.5, 140.72.1-10.a.1.6, 140.72.1-10.a.1.7, 140.72.1-10.a.1.8, 220.72.1-10.a.1.1, 220.72.1-10.a.1.2, 220.72.1-10.a.1.3, 220.72.1-10.a.1.4, 220.72.1-10.a.1.5, 220.72.1-10.a.1.6, 220.72.1-10.a.1.7, 220.72.1-10.a.1.8, 260.72.1-10.a.1.1, 260.72.1-10.a.1.2, 260.72.1-10.a.1.3, 260.72.1-10.a.1.4, 260.72.1-10.a.1.5, 260.72.1-10.a.1.6, 260.72.1-10.a.1.7, 260.72.1-10.a.1.8, 280.72.1-10.a.1.1, 280.72.1-10.a.1.2, 280.72.1-10.a.1.3, 280.72.1-10.a.1.4, 280.72.1-10.a.1.5, 280.72.1-10.a.1.6, 280.72.1-10.a.1.7, 280.72.1-10.a.1.8
Cyclic 10-isogeny field degree:	$1$
Cyclic 10-torsion field degree:	$4$
Full 10-torsion field degree:	$80$

Jacobian

Conductor:	$2^{2}\cdot5$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	20.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $

$=$

$ x^{3} + x^{2} + 4x + 4 $

Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model

$(-1:0:1)$, $(0:-2:1)$, $(4:-10:1)$, $(4:10:1)$, $(0:2:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle \frac{702x^{2}y^{10}+172801470x^{2}y^{8}z^{2}-7293296040x^{2}y^{6}z^{4}+58873460634x^{2}y^{4}z^{6}-140838145082x^{2}y^{2}z^{8}+63165448681x^{2}z^{10}+167859xy^{10}z+671160132xy^{8}z^{3}-9162527235xy^{6}z^{5}-2073715896xy^{4}z^{7}+238834250645xy^{2}z^{9}-483993059328xz^{11}+y^{12}+14328537y^{10}z^{2}-537229632y^{8}z^{4}+16163936765y^{6}z^{6}-140965016207y^{4}z^{8}+465116093315y^{2}z^{10}-516640929884z^{12}}{x^{2}y^{10}-6880x^{2}y^{8}z^{2}-27648x^{2}y^{6}z^{4}-3958272x^{2}y^{4}z^{6}-53063680x^{2}y^{2}z^{8}+33021952x^{2}z^{10}-40xy^{10}z+32720xy^{8}z^{3}+1071360xy^{6}z^{5}+11865344xy^{4}z^{7}-72558592xy^{2}z^{9}-280088576xz^{11}+700y^{10}z^{2}-101968y^{8}z^{4}-1150976y^{6}z^{6}+13327104y^{4}z^{8}+49731584y^{2}z^{10}-313110528z^{12}}$

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
$X(2)$	$2$	$6$	$6$	$0$	$0$	full Jacobian
$X_0(5)$	$5$	$6$	$6$	$0$	$0$	full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X(2)$	$2$	$6$	$6$	$0$	$0$	full Jacobian
10.12.1.a.1	$10$	$3$	$3$	$1$	$0$	dimension zero
$X_0(10)$	$10$	$2$	$2$	$0$	$0$	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
$X_{\pm1}(2,10)$	$10$	$2$	$2$	$1$	$0$	dimension zero
10.72.1.a.2	$10$	$2$	$2$	$1$	$0$	dimension zero
$X_{\mathrm{sp}}(10)$	$10$	$5$	$5$	$7$	$0$	$1^{6}$
20.72.1.a.1	$20$	$2$	$2$	$1$	$0$	dimension zero
20.72.1.a.2	$20$	$2$	$2$	$1$	$0$	dimension zero
20.72.3.a.1	$20$	$2$	$2$	$3$	$0$	$1^{2}$
20.72.3.b.1	$20$	$2$	$2$	$3$	$0$	$1^{2}$
20.72.3.c.1	$20$	$2$	$2$	$3$	$1$	$1^{2}$
20.72.3.d.1	$20$	$2$	$2$	$3$	$0$	$1^{2}$
20.72.3.e.1	$20$	$2$	$2$	$3$	$0$	$2$
20.72.3.e.2	$20$	$2$	$2$	$3$	$0$	$2$
20.72.3.f.1	$20$	$2$	$2$	$3$	$0$	$2$
20.72.3.f.2	$20$	$2$	$2$	$3$	$0$	$2$
30.72.1.b.1	$30$	$2$	$2$	$1$	$0$	dimension zero
30.72.1.b.2	$30$	$2$	$2$	$1$	$0$	dimension zero
30.108.7.a.1	$30$	$3$	$3$	$7$	$0$	$1^{6}$
30.144.7.a.1	$30$	$4$	$4$	$7$	$0$	$1^{6}$
40.72.1.a.1	$40$	$2$	$2$	$1$	$0$	dimension zero
40.72.1.a.2	$40$	$2$	$2$	$1$	$0$	dimension zero
40.72.1.b.1	$40$	$2$	$2$	$1$	$0$	dimension zero
40.72.1.b.2	$40$	$2$	$2$	$1$	$0$	dimension zero
40.72.3.a.1	$40$	$2$	$2$	$3$	$1$	$1^{2}$
40.72.3.b.1	$40$	$2$	$2$	$3$	$1$	$1^{2}$
40.72.3.c.1	$40$	$2$	$2$	$3$	$2$	$1^{2}$
40.72.3.d.1	$40$	$2$	$2$	$3$	$0$	$1^{2}$
40.72.3.e.1	$40$	$2$	$2$	$3$	$0$	$2$
40.72.3.e.2	$40$	$2$	$2$	$3$	$0$	$2$
40.72.3.f.1	$40$	$2$	$2$	$3$	$0$	$2$
40.72.3.f.2	$40$	$2$	$2$	$3$	$0$	$2$
50.180.7.a.1	$50$	$5$	$5$	$7$	$0$	$1^{6}$
60.72.1.b.1	$60$	$2$	$2$	$1$	$0$	dimension zero
60.72.1.b.2	$60$	$2$	$2$	$1$	$0$	dimension zero
60.72.3.a.1	$60$	$2$	$2$	$3$	$0$	$1^{2}$
60.72.3.b.1	$60$	$2$	$2$	$3$	$1$	$1^{2}$
60.72.3.c.1	$60$	$2$	$2$	$3$	$1$	$1^{2}$
60.72.3.d.1	$60$	$2$	$2$	$3$	$1$	$1^{2}$
60.72.3.ca.1	$60$	$2$	$2$	$3$	$0$	$2$
60.72.3.ca.2	$60$	$2$	$2$	$3$	$0$	$2$
60.72.3.cb.1	$60$	$2$	$2$	$3$	$0$	$2$
60.72.3.cb.2	$60$	$2$	$2$	$3$	$0$	$2$
70.72.1.a.1	$70$	$2$	$2$	$1$	$0$	dimension zero
70.72.1.a.2	$70$	$2$	$2$	$1$	$0$	dimension zero
70.288.19.a.1	$70$	$8$	$8$	$19$	$0$	$1^{12}\cdot2^{3}$
70.756.55.a.1	$70$	$21$	$21$	$55$	$14$	$1^{12}\cdot2^{21}$
70.1008.73.a.1	$70$	$28$	$28$	$73$	$14$	$1^{24}\cdot2^{24}$
110.72.1.a.1	$110$	$2$	$2$	$1$	$?$	dimension zero
110.72.1.a.2	$110$	$2$	$2$	$1$	$?$	dimension zero
120.72.1.c.1	$120$	$2$	$2$	$1$	$?$	dimension zero
120.72.1.c.2	$120$	$2$	$2$	$1$	$?$	dimension zero
120.72.1.d.1	$120$	$2$	$2$	$1$	$?$	dimension zero
120.72.1.d.2	$120$	$2$	$2$	$1$	$?$	dimension zero
120.72.3.a.1	$120$	$2$	$2$	$3$	$?$	not computed
120.72.3.b.1	$120$	$2$	$2$	$3$	$?$	not computed
120.72.3.c.1	$120$	$2$	$2$	$3$	$?$	not computed
120.72.3.d.1	$120$	$2$	$2$	$3$	$?$	not computed
120.72.3.fs.1	$120$	$2$	$2$	$3$	$?$	not computed
120.72.3.fs.2	$120$	$2$	$2$	$3$	$?$	not computed
120.72.3.ft.1	$120$	$2$	$2$	$3$	$?$	not computed
120.72.3.ft.2	$120$	$2$	$2$	$3$	$?$	not computed
130.72.1.a.1	$130$	$2$	$2$	$1$	$?$	dimension zero
130.72.1.a.2	$130$	$2$	$2$	$1$	$?$	dimension zero
140.72.1.a.1	$140$	$2$	$2$	$1$	$?$	dimension zero
140.72.1.a.2	$140$	$2$	$2$	$1$	$?$	dimension zero
140.72.3.a.1	$140$	$2$	$2$	$3$	$?$	not computed
140.72.3.b.1	$140$	$2$	$2$	$3$	$?$	not computed
140.72.3.c.1	$140$	$2$	$2$	$3$	$?$	not computed
140.72.3.d.1	$140$	$2$	$2$	$3$	$?$	not computed
140.72.3.e.1	$140$	$2$	$2$	$3$	$?$	not computed
140.72.3.e.2	$140$	$2$	$2$	$3$	$?$	not computed
140.72.3.f.1	$140$	$2$	$2$	$3$	$?$	not computed
140.72.3.f.2	$140$	$2$	$2$	$3$	$?$	not computed
170.72.1.a.1	$170$	$2$	$2$	$1$	$?$	dimension zero
170.72.1.a.2	$170$	$2$	$2$	$1$	$?$	dimension zero
190.72.1.a.1	$190$	$2$	$2$	$1$	$?$	dimension zero
190.72.1.a.2	$190$	$2$	$2$	$1$	$?$	dimension zero
210.72.1.a.1	$210$	$2$	$2$	$1$	$?$	dimension zero
210.72.1.a.2	$210$	$2$	$2$	$1$	$?$	dimension zero
220.72.1.a.1	$220$	$2$	$2$	$1$	$?$	dimension zero
220.72.1.a.2	$220$	$2$	$2$	$1$	$?$	dimension zero
220.72.3.a.1	$220$	$2$	$2$	$3$	$?$	not computed
220.72.3.b.1	$220$	$2$	$2$	$3$	$?$	not computed
220.72.3.c.1	$220$	$2$	$2$	$3$	$?$	not computed
220.72.3.d.1	$220$	$2$	$2$	$3$	$?$	not computed
220.72.3.e.1	$220$	$2$	$2$	$3$	$?$	not computed
220.72.3.e.2	$220$	$2$	$2$	$3$	$?$	not computed
220.72.3.f.1	$220$	$2$	$2$	$3$	$?$	not computed
220.72.3.f.2	$220$	$2$	$2$	$3$	$?$	not computed
230.72.1.a.1	$230$	$2$	$2$	$1$	$?$	dimension zero
230.72.1.a.2	$230$	$2$	$2$	$1$	$?$	dimension zero
260.72.1.a.1	$260$	$2$	$2$	$1$	$?$	dimension zero
260.72.1.a.2	$260$	$2$	$2$	$1$	$?$	dimension zero
260.72.3.a.1	$260$	$2$	$2$	$3$	$?$	not computed
260.72.3.b.1	$260$	$2$	$2$	$3$	$?$	not computed
260.72.3.c.1	$260$	$2$	$2$	$3$	$?$	not computed
260.72.3.d.1	$260$	$2$	$2$	$3$	$?$	not computed
260.72.3.e.1	$260$	$2$	$2$	$3$	$?$	not computed
260.72.3.e.2	$260$	$2$	$2$	$3$	$?$	not computed
260.72.3.f.1	$260$	$2$	$2$	$3$	$?$	not computed
260.72.3.f.2	$260$	$2$	$2$	$3$	$?$	not computed
280.72.1.a.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.72.1.a.2	$280$	$2$	$2$	$1$	$?$	dimension zero
280.72.1.b.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.72.1.b.2	$280$	$2$	$2$	$1$	$?$	dimension zero
280.72.3.a.1	$280$	$2$	$2$	$3$	$?$	not computed
280.72.3.b.1	$280$	$2$	$2$	$3$	$?$	not computed
280.72.3.c.1	$280$	$2$	$2$	$3$	$?$	not computed
280.72.3.d.1	$280$	$2$	$2$	$3$	$?$	not computed
280.72.3.e.1	$280$	$2$	$2$	$3$	$?$	not computed
280.72.3.e.2	$280$	$2$	$2$	$3$	$?$	not computed
280.72.3.f.1	$280$	$2$	$2$	$3$	$?$	not computed
280.72.3.f.2	$280$	$2$	$2$	$3$	$?$	not computed
290.72.1.a.1	$290$	$2$	$2$	$1$	$?$	dimension zero
290.72.1.a.2	$290$	$2$	$2$	$1$	$?$	dimension zero
310.72.1.a.1	$310$	$2$	$2$	$1$	$?$	dimension zero
310.72.1.a.2	$310$	$2$	$2$	$1$	$?$	dimension zero
330.72.1.a.1	$330$	$2$	$2$	$1$	$?$	dimension zero
330.72.1.a.2	$330$	$2$	$2$	$1$	$?$	dimension zero