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Properties

Label 16.24.0.n.2
Level $16$
Index $24$
Genus $0$
Analytic rank $0$
Cusps $5$
$\Q$-cusps $1$

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Invariants

Level: $16$ $\SL_2$-level: $16$
Index: $24$ $\PSL_2$-index:$24$
Genus: $0 = 1 + \frac{ 24 }{12} - \frac{ 2 }{4} - \frac{ 0 }{3} - \frac{ 5 }{2}$
Cusps: $5$ (of which $1$ is rational) Cusp widths $2^{4}\cdot16$ Cusp orbits $1\cdot4$
Elliptic points: $2$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $1$
Rational CM points: yes $\quad(D =$ $-4,-7$)

Other labels

Cummins and Pauli (CP) label: 16E0
Sutherland and Zywina (SZ) label: 16E0-16c
Rouse and Zureick-Brown (RZB) label: X112
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.24.0.36

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}5&9\\8&11\end{bmatrix}$, $\begin{bmatrix}15&4\\6&13\end{bmatrix}$, $\begin{bmatrix}15&7\\2&1\end{bmatrix}$, $\begin{bmatrix}15&14\\14&9\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 16-isogeny field degree: $8$
Cyclic 16-torsion field degree: $64$
Full 16-torsion field degree: $1024$

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 24 to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{x^{24}(x^{2}-6y^{2})^{3}(x^{2}-2y^{2})^{3}(x^{2}-2xy-2y^{2})^{3}(x^{2}+2xy-2y^{2})^{3}}{y^{16}x^{24}(x^{4}-8x^{2}y^{2}+8y^{4})^{2}}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
8.12.0.v.1 $8$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
16.48.0.c.2 $16$ $2$ $2$ $0$
16.48.0.i.1 $16$ $2$ $2$ $0$
16.48.0.q.1 $16$ $2$ $2$ $0$
16.48.0.t.2 $16$ $2$ $2$ $0$
16.48.1.ba.1 $16$ $2$ $2$ $1$
16.48.1.be.1 $16$ $2$ $2$ $1$
16.48.1.cd.1 $16$ $2$ $2$ $1$
16.48.1.ch.1 $16$ $2$ $2$ $1$
16.48.2.t.1 $16$ $2$ $2$ $2$
16.48.2.x.1 $16$ $2$ $2$ $2$
16.48.2.bq.1 $16$ $2$ $2$ $2$
16.48.2.bu.1 $16$ $2$ $2$ $2$
48.48.0.cf.2 $48$ $2$ $2$ $0$
48.48.0.ch.1 $48$ $2$ $2$ $0$
48.48.0.cn.1 $48$ $2$ $2$ $0$
48.48.0.cp.2 $48$ $2$ $2$ $0$
48.48.1.du.1 $48$ $2$ $2$ $1$
48.48.1.dy.1 $48$ $2$ $2$ $1$
48.48.1.et.1 $48$ $2$ $2$ $1$
48.48.1.ex.1 $48$ $2$ $2$ $1$
48.48.2.dn.1 $48$ $2$ $2$ $2$
48.48.2.dr.1 $48$ $2$ $2$ $2$
48.48.2.ek.1 $48$ $2$ $2$ $2$
48.48.2.eo.1 $48$ $2$ $2$ $2$
48.72.3.p.2 $48$ $3$ $3$ $3$
48.96.4.h.2 $48$ $4$ $4$ $4$
80.48.0.cn.2 $80$ $2$ $2$ $0$
80.48.0.cp.1 $80$ $2$ $2$ $0$
80.48.0.cv.1 $80$ $2$ $2$ $0$
80.48.0.cx.2 $80$ $2$ $2$ $0$
80.48.1.dw.1 $80$ $2$ $2$ $1$
80.48.1.ea.1 $80$ $2$ $2$ $1$
80.48.1.ev.1 $80$ $2$ $2$ $1$
80.48.1.ez.1 $80$ $2$ $2$ $1$
80.48.2.dx.1 $80$ $2$ $2$ $2$
80.48.2.eb.1 $80$ $2$ $2$ $2$
80.48.2.eu.1 $80$ $2$ $2$ $2$
80.48.2.ey.1 $80$ $2$ $2$ $2$
80.120.8.bj.2 $80$ $5$ $5$ $8$
80.144.7.cr.2 $80$ $6$ $6$ $7$
80.240.15.db.2 $80$ $10$ $10$ $15$
112.48.0.cf.2 $112$ $2$ $2$ $0$
112.48.0.ch.1 $112$ $2$ $2$ $0$
112.48.0.cn.1 $112$ $2$ $2$ $0$
112.48.0.cp.2 $112$ $2$ $2$ $0$
112.48.1.du.1 $112$ $2$ $2$ $1$
112.48.1.dy.1 $112$ $2$ $2$ $1$
112.48.1.et.1 $112$ $2$ $2$ $1$
112.48.1.ex.1 $112$ $2$ $2$ $1$
112.48.2.dn.1 $112$ $2$ $2$ $2$
112.48.2.dr.1 $112$ $2$ $2$ $2$
112.48.2.ek.1 $112$ $2$ $2$ $2$
112.48.2.eo.1 $112$ $2$ $2$ $2$
112.192.12.h.2 $112$ $8$ $8$ $12$
176.48.0.cf.2 $176$ $2$ $2$ $0$
176.48.0.ch.1 $176$ $2$ $2$ $0$
176.48.0.cn.1 $176$ $2$ $2$ $0$
176.48.0.cp.2 $176$ $2$ $2$ $0$
176.48.1.du.1 $176$ $2$ $2$ $1$
176.48.1.dy.1 $176$ $2$ $2$ $1$
176.48.1.et.1 $176$ $2$ $2$ $1$
176.48.1.ex.1 $176$ $2$ $2$ $1$
176.48.2.dn.1 $176$ $2$ $2$ $2$
176.48.2.dr.1 $176$ $2$ $2$ $2$
176.48.2.ek.1 $176$ $2$ $2$ $2$
176.48.2.eo.1 $176$ $2$ $2$ $2$
176.288.20.h.2 $176$ $12$ $12$ $20$
208.48.0.cn.2 $208$ $2$ $2$ $0$
208.48.0.cp.1 $208$ $2$ $2$ $0$
208.48.0.cv.1 $208$ $2$ $2$ $0$
208.48.0.cx.2 $208$ $2$ $2$ $0$
208.48.1.dw.1 $208$ $2$ $2$ $1$
208.48.1.ea.1 $208$ $2$ $2$ $1$
208.48.1.ev.1 $208$ $2$ $2$ $1$
208.48.1.ez.1 $208$ $2$ $2$ $1$
208.48.2.dx.1 $208$ $2$ $2$ $2$
208.48.2.eb.1 $208$ $2$ $2$ $2$
208.48.2.eu.1 $208$ $2$ $2$ $2$
208.48.2.ey.1 $208$ $2$ $2$ $2$
208.336.23.dx.2 $208$ $14$ $14$ $23$
240.48.0.gt.2 $240$ $2$ $2$ $0$
240.48.0.gx.1 $240$ $2$ $2$ $0$
240.48.0.hj.1 $240$ $2$ $2$ $0$
240.48.0.hn.2 $240$ $2$ $2$ $0$
240.48.1.kq.1 $240$ $2$ $2$ $1$
240.48.1.ku.1 $240$ $2$ $2$ $1$
240.48.1.mf.1 $240$ $2$ $2$ $1$
240.48.1.mj.1 $240$ $2$ $2$ $1$
240.48.2.lx.1 $240$ $2$ $2$ $2$
240.48.2.mb.1 $240$ $2$ $2$ $2$
240.48.2.nk.1 $240$ $2$ $2$ $2$
240.48.2.no.1 $240$ $2$ $2$ $2$
272.48.0.cn.1 $272$ $2$ $2$ $0$
272.48.0.cp.1 $272$ $2$ $2$ $0$
272.48.0.cv.1 $272$ $2$ $2$ $0$
272.48.0.cx.2 $272$ $2$ $2$ $0$
272.48.1.dw.1 $272$ $2$ $2$ $1$
272.48.1.ea.1 $272$ $2$ $2$ $1$
272.48.1.ev.1 $272$ $2$ $2$ $1$
272.48.1.ez.1 $272$ $2$ $2$ $1$
272.48.2.dx.1 $272$ $2$ $2$ $2$
272.48.2.eb.1 $272$ $2$ $2$ $2$
272.48.2.eu.1 $272$ $2$ $2$ $2$
272.48.2.ey.1 $272$ $2$ $2$ $2$
304.48.0.cf.2 $304$ $2$ $2$ $0$
304.48.0.ch.1 $304$ $2$ $2$ $0$
304.48.0.cn.1 $304$ $2$ $2$ $0$
304.48.0.cp.2 $304$ $2$ $2$ $0$
304.48.1.du.1 $304$ $2$ $2$ $1$
304.48.1.dy.1 $304$ $2$ $2$ $1$
304.48.1.et.1 $304$ $2$ $2$ $1$
304.48.1.ex.1 $304$ $2$ $2$ $1$
304.48.2.dn.1 $304$ $2$ $2$ $2$
304.48.2.dr.1 $304$ $2$ $2$ $2$
304.48.2.ek.1 $304$ $2$ $2$ $2$
304.48.2.eo.1 $304$ $2$ $2$ $2$