Modular curve Xsp+(7)X_{\mathrm{sp}}^+(7)

• Label: 7.28.0.a.1
• Level: $7$
• Index: $28$
• Genus: $0$
• Analytic rank: $0$
• Cusps: $4$
• Q\Q-cusps: $1$

Invariants

Level:	$7$		$\SL_2$-level:	$7$
Index:	$28$		$\PSL_2$-index:	$28$
Genus:	$0 = 1 + \frac{ 28 }{12} - \frac{ 4 }{4} - \frac{ 1 }{3} - \frac{ 4 }{2}$
Cusps:	$4$ (of which $1$ is rational)		Cusp widths	$7^{4}$		Cusp orbits	$1\cdot3$
Elliptic points:	$4$ of order $2$ and $1$ of order $3$
$\Q$-gonality:	$1$
$\overline{\Q}$-gonality:	$1$
Rational cusps:	$1$
Rational CM points:	yes $\quad(D =$ $-3,-12,-19,-27$)

Other labels

Cummins and Pauli (CP) label:	7F0
Sutherland and Zywina (SZ) label:	7F0-7a
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	7.28.0.1
Sutherland (S) label:	7Ns

Level structure

$\GL_2(\Z/7\Z)$-generators:	$\begin{bmatrix}0&3\\4&0\end{bmatrix}$, $\begin{bmatrix}1&0\\0&5\end{bmatrix}$
$\GL_2(\Z/7\Z)$-subgroup:	$C_6\wr C_2$
Contains $-I$:	yes
Quadratic refinements:	none in database
Cyclic 7-isogeny field degree:	$2$
Cyclic 7-torsion field degree:	$12$
Full 7-torsion field degree:	$72$

Models

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

Rational points

This modular curve has infinitely many rational points, including 11 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 28 to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{1}{2^7}\cdot\frac{(x-y)^{29} (x+y)^{3} (x^{2}-12 x y+15 y^{2})^{3} (x^{2}-12 x y+43 y^{2})^{3} (x^{4}-14 x^{3} y+68 x^{2} y^{2}-154 x y^{3}+211 y^{4})^{3}}{y^{7} (x-y)^{28} (x^{3}-11 x^{2} y+31 x y^{2}-13 y^{3})^{7}}$

Modular covers

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
$X(1)$	$1$	$28$	$28$	$0$	$0$

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

Covering curve	Level	Index	Degree	Genus
$X_{\mathrm{sp}}(7)$	$7$	$2$	$2$	$1$
7.56.1.b.1	$7$	$2$	$2$	$1$
7.84.1.a.1	$7$	$3$	$3$	$1$
14.56.1.a.1	$14$	$2$	$2$	$1$
14.56.1.b.1	$14$	$2$	$2$	$1$
14.56.3.a.1	$14$	$2$	$2$	$3$
14.56.3.b.1	$14$	$2$	$2$	$3$
14.84.3.a.1	$14$	$3$	$3$	$3$
21.56.1.a.1	$21$	$2$	$2$	$1$
21.56.1.b.1	$21$	$2$	$2$	$1$
21.84.3.a.1	$21$	$3$	$3$	$3$
21.112.6.a.1	$21$	$4$	$4$	$6$
28.56.1.a.1	$28$	$2$	$2$	$1$
28.56.1.b.1	$28$	$2$	$2$	$1$
28.56.1.c.1	$28$	$2$	$2$	$1$
28.56.1.d.1	$28$	$2$	$2$	$1$
28.56.3.a.1	$28$	$2$	$2$	$3$
28.56.3.b.1	$28$	$2$	$2$	$3$
28.112.6.a.1	$28$	$4$	$4$	$6$
35.56.1.a.1	$35$	$2$	$2$	$1$
35.56.1.b.1	$35$	$2$	$2$	$1$
35.140.9.a.1	$35$	$5$	$5$	$9$
35.168.9.a.1	$35$	$6$	$6$	$9$
35.280.18.a.1	$35$	$10$	$10$	$18$
42.56.1.a.1	$42$	$2$	$2$	$1$
42.56.1.b.1	$42$	$2$	$2$	$1$
42.56.3.a.1	$42$	$2$	$2$	$3$
42.56.3.b.1	$42$	$2$	$2$	$3$
49.196.9.a.1	$49$	$7$	$7$	$9$
$X_{\mathrm{sp}}^+(49)$	$49$	$49$	$49$	$94$
56.56.1.a.1	$56$	$2$	$2$	$1$
56.56.1.b.1	$56$	$2$	$2$	$1$
56.56.1.c.1	$56$	$2$	$2$	$1$
56.56.1.d.1	$56$	$2$	$2$	$1$
56.56.1.e.1	$56$	$2$	$2$	$1$
56.56.1.f.1	$56$	$2$	$2$	$1$
56.56.1.g.1	$56$	$2$	$2$	$1$
56.56.1.h.1	$56$	$2$	$2$	$1$
56.56.3.a.1	$56$	$2$	$2$	$3$
56.56.3.b.1	$56$	$2$	$2$	$3$
56.56.3.c.1	$56$	$2$	$2$	$3$
56.56.3.d.1	$56$	$2$	$2$	$3$
63.756.54.a.1	$63$	$27$	$27$	$54$
70.56.1.a.1	$70$	$2$	$2$	$1$
70.56.1.b.1	$70$	$2$	$2$	$1$
70.56.3.a.1	$70$	$2$	$2$	$3$
70.56.3.b.1	$70$	$2$	$2$	$3$
77.56.1.a.1	$77$	$2$	$2$	$1$
77.56.1.b.1	$77$	$2$	$2$	$1$
84.56.1.a.1	$84$	$2$	$2$	$1$
84.56.1.b.1	$84$	$2$	$2$	$1$
84.56.1.c.1	$84$	$2$	$2$	$1$
84.56.1.d.1	$84$	$2$	$2$	$1$
84.56.3.a.1	$84$	$2$	$2$	$3$
84.56.3.b.1	$84$	$2$	$2$	$3$
91.56.1.a.1	$91$	$2$	$2$	$1$
91.56.1.b.1	$91$	$2$	$2$	$1$
105.56.1.a.1	$105$	$2$	$2$	$1$
105.56.1.b.1	$105$	$2$	$2$	$1$
119.56.1.a.1	$119$	$2$	$2$	$1$
119.56.1.b.1	$119$	$2$	$2$	$1$
133.56.1.a.1	$133$	$2$	$2$	$1$
133.56.1.b.1	$133$	$2$	$2$	$1$
140.56.1.a.1	$140$	$2$	$2$	$1$
140.56.1.b.1	$140$	$2$	$2$	$1$
140.56.1.c.1	$140$	$2$	$2$	$1$
140.56.1.d.1	$140$	$2$	$2$	$1$
140.56.3.a.1	$140$	$2$	$2$	$3$
140.56.3.b.1	$140$	$2$	$2$	$3$
154.56.1.a.1	$154$	$2$	$2$	$1$
154.56.1.b.1	$154$	$2$	$2$	$1$
154.56.3.a.1	$154$	$2$	$2$	$3$
154.56.3.b.1	$154$	$2$	$2$	$3$
161.56.1.a.1	$161$	$2$	$2$	$1$
161.56.1.b.1	$161$	$2$	$2$	$1$
168.56.1.a.1	$168$	$2$	$2$	$1$
168.56.1.b.1	$168$	$2$	$2$	$1$
168.56.1.c.1	$168$	$2$	$2$	$1$
168.56.1.d.1	$168$	$2$	$2$	$1$
168.56.1.e.1	$168$	$2$	$2$	$1$
168.56.1.f.1	$168$	$2$	$2$	$1$
168.56.1.g.1	$168$	$2$	$2$	$1$
168.56.1.h.1	$168$	$2$	$2$	$1$
168.56.3.a.1	$168$	$2$	$2$	$3$
168.56.3.b.1	$168$	$2$	$2$	$3$
168.56.3.c.1	$168$	$2$	$2$	$3$
168.56.3.d.1	$168$	$2$	$2$	$3$
182.56.1.g.1	$182$	$2$	$2$	$1$
182.56.1.h.1	$182$	$2$	$2$	$1$
182.56.3.a.1	$182$	$2$	$2$	$3$
182.56.3.b.1	$182$	$2$	$2$	$3$
203.56.1.a.1	$203$	$2$	$2$	$1$
203.56.1.b.1	$203$	$2$	$2$	$1$
210.56.1.a.1	$210$	$2$	$2$	$1$
210.56.1.b.1	$210$	$2$	$2$	$1$
210.56.3.a.1	$210$	$2$	$2$	$3$
210.56.3.b.1	$210$	$2$	$2$	$3$
217.56.1.a.1	$217$	$2$	$2$	$1$
217.56.1.b.1	$217$	$2$	$2$	$1$
231.56.1.a.1	$231$	$2$	$2$	$1$
231.56.1.b.1	$231$	$2$	$2$	$1$
238.56.1.a.1	$238$	$2$	$2$	$1$
238.56.1.b.1	$238$	$2$	$2$	$1$
238.56.3.a.1	$238$	$2$	$2$	$3$
238.56.3.b.1	$238$	$2$	$2$	$3$
259.56.1.a.1	$259$	$2$	$2$	$1$
259.56.1.b.1	$259$	$2$	$2$	$1$
266.56.1.a.1	$266$	$2$	$2$	$1$
266.56.1.b.1	$266$	$2$	$2$	$1$
266.56.3.a.1	$266$	$2$	$2$	$3$
266.56.3.b.1	$266$	$2$	$2$	$3$
273.56.1.a.1	$273$	$2$	$2$	$1$
273.56.1.b.1	$273$	$2$	$2$	$1$
280.56.1.a.1	$280$	$2$	$2$	$1$
280.56.1.b.1	$280$	$2$	$2$	$1$
280.56.1.c.1	$280$	$2$	$2$	$1$
280.56.1.d.1	$280$	$2$	$2$	$1$
280.56.1.e.1	$280$	$2$	$2$	$1$
280.56.1.f.1	$280$	$2$	$2$	$1$
280.56.1.g.1	$280$	$2$	$2$	$1$
280.56.1.h.1	$280$	$2$	$2$	$1$
280.56.3.a.1	$280$	$2$	$2$	$3$
280.56.3.b.1	$280$	$2$	$2$	$3$
280.56.3.c.1	$280$	$2$	$2$	$3$
280.56.3.d.1	$280$	$2$	$2$	$3$
287.56.1.a.1	$287$	$2$	$2$	$1$
287.56.1.b.1	$287$	$2$	$2$	$1$
301.56.1.a.1	$301$	$2$	$2$	$1$
301.56.1.b.1	$301$	$2$	$2$	$1$
308.56.1.a.1	$308$	$2$	$2$	$1$
308.56.1.b.1	$308$	$2$	$2$	$1$
308.56.1.c.1	$308$	$2$	$2$	$1$
308.56.1.d.1	$308$	$2$	$2$	$1$
308.56.3.a.1	$308$	$2$	$2$	$3$
308.56.3.b.1	$308$	$2$	$2$	$3$
322.56.1.a.1	$322$	$2$	$2$	$1$
322.56.1.b.1	$322$	$2$	$2$	$1$
322.56.3.a.1	$322$	$2$	$2$	$3$
322.56.3.b.1	$322$	$2$	$2$	$3$
329.56.1.a.1	$329$	$2$	$2$	$1$
329.56.1.b.1	$329$	$2$	$2$	$1$