Modular curve 28.12.0.g.1

• Label: 28.12.0.g.1
• Level: $28$
• Index: $12$
• Genus: $0$
• Analytic rank: $0$
• Cusps: $4$
• $\Q$-cusps: $2$

Invariants

Level:	$28$		$\SL_2$-level:	$4$
Index:	$12$		$\PSL_2$-index:	$12$
Genus:	$0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps:	$4$ (of which $2$ are rational)		Cusp widths	$2^{2}\cdot4^{2}$		Cusp orbits	$1^{2}\cdot2$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
$\Q$-gonality:	$1$
$\overline{\Q}$-gonality:	$1$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	4E0
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	28.12.0.9

Level structure

$\GL_2(\Z/28\Z)$-generators:	$\begin{bmatrix}7&22\\20&21\end{bmatrix}$, $\begin{bmatrix}11&16\\2&5\end{bmatrix}$, $\begin{bmatrix}26&5\\5&2\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	28.24.0-28.g.1.1, 28.24.0-28.g.1.2, 56.24.0-28.g.1.1, 56.24.0-28.g.1.2, 56.24.0-28.g.1.3, 56.24.0-28.g.1.4, 56.24.0-28.g.1.5, 56.24.0-28.g.1.6, 84.24.0-28.g.1.1, 84.24.0-28.g.1.2, 140.24.0-28.g.1.1, 140.24.0-28.g.1.2, 168.24.0-28.g.1.1, 168.24.0-28.g.1.2, 168.24.0-28.g.1.3, 168.24.0-28.g.1.4, 168.24.0-28.g.1.5, 168.24.0-28.g.1.6, 280.24.0-28.g.1.1, 280.24.0-28.g.1.2, 280.24.0-28.g.1.3, 280.24.0-28.g.1.4, 280.24.0-28.g.1.5, 280.24.0-28.g.1.6, 308.24.0-28.g.1.1, 308.24.0-28.g.1.2
Cyclic 28-isogeny field degree:	$8$
Cyclic 28-torsion field degree:	$96$
Full 28-torsion field degree:	$16128$

Models

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

Rational points

This modular curve has infinitely many rational points, including 631 stored non-cuspidal points.

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle \frac{2^4}{7}\cdot\frac{(x+8y)^{12}(3x^{4}-784x^{2}y^{2}-6272xy^{3}-12544y^{4})^{3}}{x^{2}(x+8y)^{14}(x^{2}+14xy+56y^{2})^{4}}$

Modular covers

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
$X_0(4)$	$4$	$2$	$2$	$0$	$0$
14.6.0.b.1	$14$	$2$	$2$	$0$	$0$
28.6.0.d.1	$28$	$2$	$2$	$0$	$0$

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

Covering curve	Level	Index	Degree	Genus
28.96.5.k.1	$28$	$8$	$8$	$5$
28.252.16.s.1	$28$	$21$	$21$	$16$
28.336.21.s.1	$28$	$28$	$28$	$21$
56.24.0.be.1	$56$	$2$	$2$	$0$
56.24.0.bf.1	$56$	$2$	$2$	$0$
56.24.0.bm.1	$56$	$2$	$2$	$0$
56.24.0.bn.1	$56$	$2$	$2$	$0$
84.36.2.s.1	$84$	$3$	$3$	$2$
84.48.1.k.1	$84$	$4$	$4$	$1$
140.60.4.k.1	$140$	$5$	$5$	$4$
140.72.3.o.1	$140$	$6$	$6$	$3$
140.120.7.s.1	$140$	$10$	$10$	$7$
168.24.0.cg.1	$168$	$2$	$2$	$0$
168.24.0.ch.1	$168$	$2$	$2$	$0$
168.24.0.co.1	$168$	$2$	$2$	$0$
168.24.0.cp.1	$168$	$2$	$2$	$0$
252.324.22.ba.1	$252$	$27$	$27$	$22$
280.24.0.cm.1	$280$	$2$	$2$	$0$
280.24.0.cn.1	$280$	$2$	$2$	$0$
280.24.0.cu.1	$280$	$2$	$2$	$0$
280.24.0.cv.1	$280$	$2$	$2$	$0$
308.144.9.k.1	$308$	$12$	$12$	$9$