Modular curve 10.12.1.a.1

• Label: 10.12.1.a.1
• Level: $10$
• Index: $12$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $2$
• $\Q$-cusps: $2$

Invariants

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

Other labels

Cummins and Pauli (CP) label:	10A1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	10.12.1.1

Level structure

$\GL_2(\Z/10\Z)$-generators:	$\begin{bmatrix}7&3\\7&2\end{bmatrix}$, $\begin{bmatrix}9&0\\4&7\end{bmatrix}$
$\GL_2(\Z/10\Z)$-subgroup:	$C_{12}\times F_5$
Contains $-I$:	yes
Quadratic refinements:	20.24.1-10.a.1.1, 20.24.1-10.a.1.2, 20.24.1-10.a.1.3, 20.24.1-10.a.1.4, 40.24.1-10.a.1.1, 40.24.1-10.a.1.2, 40.24.1-10.a.1.3, 40.24.1-10.a.1.4, 60.24.1-10.a.1.1, 60.24.1-10.a.1.2, 60.24.1-10.a.1.3, 60.24.1-10.a.1.4, 120.24.1-10.a.1.1, 120.24.1-10.a.1.2, 120.24.1-10.a.1.3, 120.24.1-10.a.1.4, 140.24.1-10.a.1.1, 140.24.1-10.a.1.2, 140.24.1-10.a.1.3, 140.24.1-10.a.1.4, 220.24.1-10.a.1.1, 220.24.1-10.a.1.2, 220.24.1-10.a.1.3, 220.24.1-10.a.1.4, 260.24.1-10.a.1.1, 260.24.1-10.a.1.2, 260.24.1-10.a.1.3, 260.24.1-10.a.1.4, 280.24.1-10.a.1.1, 280.24.1-10.a.1.2, 280.24.1-10.a.1.3, 280.24.1-10.a.1.4
Cyclic 10-isogeny field degree:	$3$
Cyclic 10-torsion field degree:	$12$
Full 10-torsion field degree:	$240$

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} - 36x - 140 $

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model

$(0:1:0)$, $(7:0:1)$

Maps to other modular curves

$j$-invariant map of degree 12 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -\frac{14x^{2}y^{2}-x^{2}z^{2}-85xy^{2}z-1736xz^{3}-y^{4}-83y^{2}z^{2}+12076z^{4}}{z^{3}(x-7z)}$

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
$X_{\mathrm{ns}}(2)$	$2$	$6$	$6$	$0$	$0$	full Jacobian
$X_0(5)$	$5$	$2$	$2$	$0$	$0$	full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X_{\mathrm{ns}}(2)$	$2$	$6$	$6$	$0$	$0$	full Jacobian
$X_0(5)$	$5$	$2$	$2$	$0$	$0$	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
10.24.1.a.1	$10$	$2$	$2$	$1$	$0$	dimension zero
10.24.1.a.2	$10$	$2$	$2$	$1$	$0$	dimension zero
10.36.1.a.1	$10$	$3$	$3$	$1$	$0$	dimension zero
10.60.3.b.1	$10$	$5$	$5$	$3$	$0$	$1^{2}$
20.24.1.a.1	$20$	$2$	$2$	$1$	$0$	dimension zero
20.24.1.a.2	$20$	$2$	$2$	$1$	$0$	dimension zero
20.48.3.e.1	$20$	$4$	$4$	$3$	$0$	$1^{2}$
30.24.1.b.1	$30$	$2$	$2$	$1$	$0$	dimension zero
30.24.1.b.2	$30$	$2$	$2$	$1$	$0$	dimension zero
30.36.3.a.1	$30$	$3$	$3$	$3$	$0$	$1^{2}$
30.48.3.a.1	$30$	$4$	$4$	$3$	$0$	$1^{2}$
40.24.1.bw.1	$40$	$2$	$2$	$1$	$0$	dimension zero
40.24.1.bw.2	$40$	$2$	$2$	$1$	$0$	dimension zero
40.24.1.bz.1	$40$	$2$	$2$	$1$	$0$	dimension zero
40.24.1.bz.2	$40$	$2$	$2$	$1$	$0$	dimension zero
50.60.3.a.1	$50$	$5$	$5$	$3$	$0$	$1^{2}$
60.24.1.d.1	$60$	$2$	$2$	$1$	$0$	dimension zero
60.24.1.d.2	$60$	$2$	$2$	$1$	$0$	dimension zero
70.24.1.a.1	$70$	$2$	$2$	$1$	$0$	dimension zero
70.24.1.a.2	$70$	$2$	$2$	$1$	$0$	dimension zero
70.36.1.a.1	$70$	$3$	$3$	$1$	$0$	dimension zero
70.96.7.a.1	$70$	$8$	$8$	$7$	$0$	$1^{4}\cdot2$
70.252.19.a.1	$70$	$21$	$21$	$19$	$4$	$1^{4}\cdot2^{7}$
70.336.25.a.1	$70$	$28$	$28$	$25$	$4$	$1^{8}\cdot2^{8}$
90.36.1.b.1	$90$	$3$	$3$	$1$	$?$	dimension zero
110.24.1.a.1	$110$	$2$	$2$	$1$	$?$	dimension zero
110.24.1.a.2	$110$	$2$	$2$	$1$	$?$	dimension zero
110.144.11.a.1	$110$	$12$	$12$	$11$	$?$	not computed
120.24.1.ci.1	$120$	$2$	$2$	$1$	$?$	dimension zero
120.24.1.ci.2	$120$	$2$	$2$	$1$	$?$	dimension zero
120.24.1.cl.1	$120$	$2$	$2$	$1$	$?$	dimension zero
120.24.1.cl.2	$120$	$2$	$2$	$1$	$?$	dimension zero
130.24.1.a.1	$130$	$2$	$2$	$1$	$?$	dimension zero
130.24.1.a.2	$130$	$2$	$2$	$1$	$?$	dimension zero
130.36.1.a.1	$130$	$3$	$3$	$1$	$?$	dimension zero
130.168.13.a.1	$130$	$14$	$14$	$13$	$?$	not computed
140.24.1.a.1	$140$	$2$	$2$	$1$	$?$	dimension zero
140.24.1.a.2	$140$	$2$	$2$	$1$	$?$	dimension zero
170.24.1.a.1	$170$	$2$	$2$	$1$	$?$	dimension zero
170.24.1.a.2	$170$	$2$	$2$	$1$	$?$	dimension zero
170.216.17.a.1	$170$	$18$	$18$	$17$	$?$	not computed
190.24.1.a.1	$190$	$2$	$2$	$1$	$?$	dimension zero
190.24.1.a.2	$190$	$2$	$2$	$1$	$?$	dimension zero
190.36.1.a.1	$190$	$3$	$3$	$1$	$?$	dimension zero
190.240.19.a.1	$190$	$20$	$20$	$19$	$?$	not computed
210.24.1.a.1	$210$	$2$	$2$	$1$	$?$	dimension zero
210.24.1.a.2	$210$	$2$	$2$	$1$	$?$	dimension zero
220.24.1.a.1	$220$	$2$	$2$	$1$	$?$	dimension zero
220.24.1.a.2	$220$	$2$	$2$	$1$	$?$	dimension zero
230.24.1.a.1	$230$	$2$	$2$	$1$	$?$	dimension zero
230.24.1.a.2	$230$	$2$	$2$	$1$	$?$	dimension zero
230.288.23.a.1	$230$	$24$	$24$	$23$	$?$	not computed
260.24.1.a.1	$260$	$2$	$2$	$1$	$?$	dimension zero
260.24.1.a.2	$260$	$2$	$2$	$1$	$?$	dimension zero
280.24.1.bw.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.24.1.bw.2	$280$	$2$	$2$	$1$	$?$	dimension zero
280.24.1.bz.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.24.1.bz.2	$280$	$2$	$2$	$1$	$?$	dimension zero
290.24.1.a.1	$290$	$2$	$2$	$1$	$?$	dimension zero
290.24.1.a.2	$290$	$2$	$2$	$1$	$?$	dimension zero
310.24.1.a.1	$310$	$2$	$2$	$1$	$?$	dimension zero
310.24.1.a.2	$310$	$2$	$2$	$1$	$?$	dimension zero
310.36.1.a.1	$310$	$3$	$3$	$1$	$?$	dimension zero
330.24.1.a.1	$330$	$2$	$2$	$1$	$?$	dimension zero
330.24.1.a.2	$330$	$2$	$2$	$1$	$?$	dimension zero