Modular curve 24.24.0.i.2

• Label: 24.24.0.i.2
• Level: $24$
• Index: $24$
• Genus: $0$
• Analytic rank: $0$
• Cusps: $6$
• $\Q$-cusps: $2$

Invariants

Level:	$24$		$\SL_2$-level:	$8$
Index:	$24$		$\PSL_2$-index:	$24$
Genus:	$0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps:	$6$ (of which $2$ are rational)		Cusp widths	$2^{2}\cdot4^{3}\cdot8$		Cusp orbits	$1^{2}\cdot2^{2}$
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:	8J0
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	24.24.0.13

Level structure

$\GL_2(\Z/24\Z)$-generators:	$\begin{bmatrix}1&20\\4&1\end{bmatrix}$, $\begin{bmatrix}9&16\\8&5\end{bmatrix}$, $\begin{bmatrix}11&14\\0&13\end{bmatrix}$, $\begin{bmatrix}13&10\\8&21\end{bmatrix}$, $\begin{bmatrix}13&16\\16&23\end{bmatrix}$, $\begin{bmatrix}21&10\\20&23\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	24.48.0-24.i.2.1, 24.48.0-24.i.2.2, 24.48.0-24.i.2.3, 24.48.0-24.i.2.4, 24.48.0-24.i.2.5, 24.48.0-24.i.2.6, 24.48.0-24.i.2.7, 24.48.0-24.i.2.8, 24.48.0-24.i.2.9, 24.48.0-24.i.2.10, 24.48.0-24.i.2.11, 24.48.0-24.i.2.12, 24.48.0-24.i.2.13, 24.48.0-24.i.2.14, 24.48.0-24.i.2.15, 24.48.0-24.i.2.16, 24.48.0-24.i.2.17, 24.48.0-24.i.2.18, 24.48.0-24.i.2.19, 24.48.0-24.i.2.20, 24.48.0-24.i.2.21, 24.48.0-24.i.2.22, 24.48.0-24.i.2.23, 24.48.0-24.i.2.24, 24.48.0-24.i.2.25, 24.48.0-24.i.2.26, 24.48.0-24.i.2.27, 24.48.0-24.i.2.28, 24.48.0-24.i.2.29, 24.48.0-24.i.2.30, 24.48.0-24.i.2.31, 24.48.0-24.i.2.32, 120.48.0-24.i.2.1, 120.48.0-24.i.2.2, 120.48.0-24.i.2.3, 120.48.0-24.i.2.4, 120.48.0-24.i.2.5, 120.48.0-24.i.2.6, 120.48.0-24.i.2.7, 120.48.0-24.i.2.8, 120.48.0-24.i.2.9, 120.48.0-24.i.2.10, 120.48.0-24.i.2.11, 120.48.0-24.i.2.12, 120.48.0-24.i.2.13, 120.48.0-24.i.2.14, 120.48.0-24.i.2.15, 120.48.0-24.i.2.16, 120.48.0-24.i.2.17, 120.48.0-24.i.2.18, 120.48.0-24.i.2.19, 120.48.0-24.i.2.20, 120.48.0-24.i.2.21, 120.48.0-24.i.2.22, 120.48.0-24.i.2.23, 120.48.0-24.i.2.24, 120.48.0-24.i.2.25, 120.48.0-24.i.2.26, 120.48.0-24.i.2.27, 120.48.0-24.i.2.28, 120.48.0-24.i.2.29, 120.48.0-24.i.2.30, 120.48.0-24.i.2.31, 120.48.0-24.i.2.32, 168.48.0-24.i.2.1, 168.48.0-24.i.2.2, 168.48.0-24.i.2.3, 168.48.0-24.i.2.4, 168.48.0-24.i.2.5, 168.48.0-24.i.2.6, 168.48.0-24.i.2.7, 168.48.0-24.i.2.8, 168.48.0-24.i.2.9, 168.48.0-24.i.2.10, 168.48.0-24.i.2.11, 168.48.0-24.i.2.12, 168.48.0-24.i.2.13, 168.48.0-24.i.2.14, 168.48.0-24.i.2.15, 168.48.0-24.i.2.16, 168.48.0-24.i.2.17, 168.48.0-24.i.2.18, 168.48.0-24.i.2.19, 168.48.0-24.i.2.20, 168.48.0-24.i.2.21, 168.48.0-24.i.2.22, 168.48.0-24.i.2.23, 168.48.0-24.i.2.24, 168.48.0-24.i.2.25, 168.48.0-24.i.2.26, 168.48.0-24.i.2.27, 168.48.0-24.i.2.28, 168.48.0-24.i.2.29, 168.48.0-24.i.2.30, 168.48.0-24.i.2.31, 168.48.0-24.i.2.32, 264.48.0-24.i.2.1, 264.48.0-24.i.2.2, 264.48.0-24.i.2.3, 264.48.0-24.i.2.4, 264.48.0-24.i.2.5, 264.48.0-24.i.2.6, 264.48.0-24.i.2.7, 264.48.0-24.i.2.8, 264.48.0-24.i.2.9, 264.48.0-24.i.2.10, 264.48.0-24.i.2.11, 264.48.0-24.i.2.12, 264.48.0-24.i.2.13, 264.48.0-24.i.2.14, 264.48.0-24.i.2.15, 264.48.0-24.i.2.16, 264.48.0-24.i.2.17, 264.48.0-24.i.2.18, 264.48.0-24.i.2.19, 264.48.0-24.i.2.20, 264.48.0-24.i.2.21, 264.48.0-24.i.2.22, 264.48.0-24.i.2.23, 264.48.0-24.i.2.24, 264.48.0-24.i.2.25, 264.48.0-24.i.2.26, 264.48.0-24.i.2.27, 264.48.0-24.i.2.28, 264.48.0-24.i.2.29, 264.48.0-24.i.2.30, 264.48.0-24.i.2.31, 264.48.0-24.i.2.32, 312.48.0-24.i.2.1, 312.48.0-24.i.2.2, 312.48.0-24.i.2.3, 312.48.0-24.i.2.4, 312.48.0-24.i.2.5, 312.48.0-24.i.2.6, 312.48.0-24.i.2.7, 312.48.0-24.i.2.8, 312.48.0-24.i.2.9, 312.48.0-24.i.2.10, 312.48.0-24.i.2.11, 312.48.0-24.i.2.12, 312.48.0-24.i.2.13, 312.48.0-24.i.2.14, 312.48.0-24.i.2.15, 312.48.0-24.i.2.16, 312.48.0-24.i.2.17, 312.48.0-24.i.2.18, 312.48.0-24.i.2.19, 312.48.0-24.i.2.20, 312.48.0-24.i.2.21, 312.48.0-24.i.2.22, 312.48.0-24.i.2.23, 312.48.0-24.i.2.24, 312.48.0-24.i.2.25, 312.48.0-24.i.2.26, 312.48.0-24.i.2.27, 312.48.0-24.i.2.28, 312.48.0-24.i.2.29, 312.48.0-24.i.2.30, 312.48.0-24.i.2.31, 312.48.0-24.i.2.32
Cyclic 24-isogeny field degree:	$8$
Cyclic 24-torsion field degree:	$64$
Full 24-torsion field degree:	$3072$

Models

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

Rational points

This modular curve has infinitely many rational points, including 90 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 24 to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{1}{2^6\cdot3^4}\cdot\frac{x^{24}(81x^{8}+3456x^{6}y^{2}+184320x^{4}y^{4}+3145728x^{2}y^{6}+16777216y^{8})^{3}}{y^{4}x^{32}(3x^{2}+32y^{2})^{2}(3x^{2}+64y^{2})^{4}}$

Modular covers

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
$X_{\pm1}(2,4)$	$4$	$2$	$2$	$0$	$0$

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

Covering curve	Level	Index	Degree	Genus
24.48.0.b.1	$24$	$2$	$2$	$0$
24.48.0.c.1	$24$	$2$	$2$	$0$
24.48.0.e.2	$24$	$2$	$2$	$0$
24.48.0.f.1	$24$	$2$	$2$	$0$
24.48.0.h.2	$24$	$2$	$2$	$0$
24.48.0.j.1	$24$	$2$	$2$	$0$
24.48.0.l.2	$24$	$2$	$2$	$0$
24.48.0.n.2	$24$	$2$	$2$	$0$
24.48.0.q.1	$24$	$2$	$2$	$0$
24.48.0.s.1	$24$	$2$	$2$	$0$
24.48.0.u.2	$24$	$2$	$2$	$0$
24.48.0.w.2	$24$	$2$	$2$	$0$
24.48.0.y.1	$24$	$2$	$2$	$0$
24.48.0.z.1	$24$	$2$	$2$	$0$
24.48.0.bb.2	$24$	$2$	$2$	$0$
24.48.0.bc.2	$24$	$2$	$2$	$0$
24.48.1.q.2	$24$	$2$	$2$	$1$
24.48.1.s.2	$24$	$2$	$2$	$1$
24.48.1.x.2	$24$	$2$	$2$	$1$
24.48.1.y.1	$24$	$2$	$2$	$1$
24.48.1.bd.2	$24$	$2$	$2$	$1$
24.48.1.bf.1	$24$	$2$	$2$	$1$
24.48.1.bh.2	$24$	$2$	$2$	$1$
24.48.1.bj.2	$24$	$2$	$2$	$1$
24.72.4.y.2	$24$	$3$	$3$	$4$
24.96.3.bp.1	$24$	$4$	$4$	$3$
120.48.0.t.1	$120$	$2$	$2$	$0$
120.48.0.u.1	$120$	$2$	$2$	$0$
120.48.0.x.1	$120$	$2$	$2$	$0$
120.48.0.y.2	$120$	$2$	$2$	$0$
120.48.0.bd.1	$120$	$2$	$2$	$0$
120.48.0.bf.1	$120$	$2$	$2$	$0$
120.48.0.bl.1	$120$	$2$	$2$	$0$
120.48.0.bn.2	$120$	$2$	$2$	$0$
120.48.0.bt.1	$120$	$2$	$2$	$0$
120.48.0.bv.1	$120$	$2$	$2$	$0$
120.48.0.cb.2	$120$	$2$	$2$	$0$
120.48.0.cd.1	$120$	$2$	$2$	$0$
120.48.0.ch.1	$120$	$2$	$2$	$0$
120.48.0.ci.1	$120$	$2$	$2$	$0$
120.48.0.cl.2	$120$	$2$	$2$	$0$
120.48.0.cm.2	$120$	$2$	$2$	$0$
120.48.1.dm.2	$120$	$2$	$2$	$1$
120.48.1.dn.1	$120$	$2$	$2$	$1$
120.48.1.dq.1	$120$	$2$	$2$	$1$
120.48.1.dr.2	$120$	$2$	$2$	$1$
120.48.1.dw.2	$120$	$2$	$2$	$1$
120.48.1.dy.2	$120$	$2$	$2$	$1$
120.48.1.ee.2	$120$	$2$	$2$	$1$
120.48.1.eg.2	$120$	$2$	$2$	$1$
120.120.8.bd.1	$120$	$5$	$5$	$8$
120.144.7.ys.1	$120$	$6$	$6$	$7$
120.240.15.bp.1	$120$	$10$	$10$	$15$
168.48.0.r.1	$168$	$2$	$2$	$0$
168.48.0.s.2	$168$	$2$	$2$	$0$
168.48.0.v.2	$168$	$2$	$2$	$0$
168.48.0.w.2	$168$	$2$	$2$	$0$
168.48.0.bb.2	$168$	$2$	$2$	$0$
168.48.0.bd.1	$168$	$2$	$2$	$0$
168.48.0.bj.1	$168$	$2$	$2$	$0$
168.48.0.bl.2	$168$	$2$	$2$	$0$
168.48.0.br.1	$168$	$2$	$2$	$0$
168.48.0.bt.2	$168$	$2$	$2$	$0$
168.48.0.bz.1	$168$	$2$	$2$	$0$
168.48.0.cb.2	$168$	$2$	$2$	$0$
168.48.0.cf.2	$168$	$2$	$2$	$0$
168.48.0.cg.1	$168$	$2$	$2$	$0$
168.48.0.cj.2	$168$	$2$	$2$	$0$
168.48.0.ck.1	$168$	$2$	$2$	$0$
168.48.1.dm.2	$168$	$2$	$2$	$1$
168.48.1.dn.2	$168$	$2$	$2$	$1$
168.48.1.dq.1	$168$	$2$	$2$	$1$
168.48.1.dr.2	$168$	$2$	$2$	$1$
168.48.1.dw.2	$168$	$2$	$2$	$1$
168.48.1.dy.2	$168$	$2$	$2$	$1$
168.48.1.ee.2	$168$	$2$	$2$	$1$
168.48.1.eg.2	$168$	$2$	$2$	$1$
168.192.11.bl.2	$168$	$8$	$8$	$11$
264.48.0.r.1	$264$	$2$	$2$	$0$
264.48.0.s.2	$264$	$2$	$2$	$0$
264.48.0.v.2	$264$	$2$	$2$	$0$
264.48.0.w.2	$264$	$2$	$2$	$0$
264.48.0.bb.2	$264$	$2$	$2$	$0$
264.48.0.bd.1	$264$	$2$	$2$	$0$
264.48.0.bj.2	$264$	$2$	$2$	$0$
264.48.0.bl.2	$264$	$2$	$2$	$0$
264.48.0.br.1	$264$	$2$	$2$	$0$
264.48.0.bt.1	$264$	$2$	$2$	$0$
264.48.0.bz.2	$264$	$2$	$2$	$0$
264.48.0.cb.1	$264$	$2$	$2$	$0$
264.48.0.cf.1	$264$	$2$	$2$	$0$
264.48.0.cg.1	$264$	$2$	$2$	$0$
264.48.0.cj.1	$264$	$2$	$2$	$0$
264.48.0.ck.2	$264$	$2$	$2$	$0$
264.48.1.dm.2	$264$	$2$	$2$	$1$
264.48.1.dn.2	$264$	$2$	$2$	$1$
264.48.1.dq.2	$264$	$2$	$2$	$1$
264.48.1.dr.2	$264$	$2$	$2$	$1$
264.48.1.dw.2	$264$	$2$	$2$	$1$
264.48.1.dy.2	$264$	$2$	$2$	$1$
264.48.1.ee.2	$264$	$2$	$2$	$1$
264.48.1.eg.2	$264$	$2$	$2$	$1$
264.288.19.sj.2	$264$	$12$	$12$	$19$
312.48.0.t.1	$312$	$2$	$2$	$0$
312.48.0.u.2	$312$	$2$	$2$	$0$
312.48.0.x.2	$312$	$2$	$2$	$0$
312.48.0.y.2	$312$	$2$	$2$	$0$
312.48.0.bd.2	$312$	$2$	$2$	$0$
312.48.0.bf.1	$312$	$2$	$2$	$0$
312.48.0.bl.1	$312$	$2$	$2$	$0$
312.48.0.bn.2	$312$	$2$	$2$	$0$
312.48.0.bt.1	$312$	$2$	$2$	$0$
312.48.0.bv.2	$312$	$2$	$2$	$0$
312.48.0.cb.1	$312$	$2$	$2$	$0$
312.48.0.cd.2	$312$	$2$	$2$	$0$
312.48.0.ch.2	$312$	$2$	$2$	$0$
312.48.0.ci.1	$312$	$2$	$2$	$0$
312.48.0.cl.2	$312$	$2$	$2$	$0$
312.48.0.cm.1	$312$	$2$	$2$	$0$
312.48.1.dm.2	$312$	$2$	$2$	$1$
312.48.1.dn.2	$312$	$2$	$2$	$1$
312.48.1.dq.2	$312$	$2$	$2$	$1$
312.48.1.dr.2	$312$	$2$	$2$	$1$
312.48.1.dw.2	$312$	$2$	$2$	$1$
312.48.1.dy.2	$312$	$2$	$2$	$1$
312.48.1.ee.2	$312$	$2$	$2$	$1$
312.48.1.eg.2	$312$	$2$	$2$	$1$
312.336.23.bl.1	$312$	$14$	$14$	$23$