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Properties

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

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Invariants

Level: $16$ $\SL_2$-level: $16$
Index: $24$ $\PSL_2$-index:$24$
Genus: $0 = 1 + \frac{ 24 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$
Cusps: $2$ (all of which are rational) Cusp widths $8\cdot16$ Cusp orbits $1^{2}$
Elliptic points: $8$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $2$
Rational CM points: yes $\quad(D =$ $-7,-12$)

Other labels

Cummins and Pauli (CP) label: 16B0
Sutherland and Zywina (SZ) label: 16B0-16a
Rouse and Zureick-Brown (RZB) label: X113
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.24.0.39

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}3&5\\10&5\end{bmatrix}$, $\begin{bmatrix}3&10\\4&3\end{bmatrix}$, $\begin{bmatrix}9&12\\14&7\end{bmatrix}$, $\begin{bmatrix}11&14\\0&7\end{bmatrix}$
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 3 stored non-cuspidal points.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{x^{24}(x^{2}-2y^{2})^{3}(x^{2}+2y^{2})^{3}(x^{2}-2xy+2y^{2})^{3}(x^{2}+2xy+2y^{2})^{3}}{y^{16}x^{32}}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
8.12.0.z.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
16.48.1.cn.1 $16$ $2$ $2$ $1$
16.48.1.co.1 $16$ $2$ $2$ $1$
16.48.1.cq.1 $16$ $2$ $2$ $1$
16.48.1.ct.1 $16$ $2$ $2$ $1$
16.48.1.cz.1 $16$ $2$ $2$ $1$
16.48.1.da.1 $16$ $2$ $2$ $1$
16.48.1.dc.1 $16$ $2$ $2$ $1$
16.48.1.df.1 $16$ $2$ $2$ $1$
16.48.3.a.2 $16$ $2$ $2$ $3$
16.48.3.m.2 $16$ $2$ $2$ $3$
16.48.3.bc.2 $16$ $2$ $2$ $3$
16.48.3.bf.1 $16$ $2$ $2$ $3$
48.48.1.iv.1 $48$ $2$ $2$ $1$
48.48.1.iw.1 $48$ $2$ $2$ $1$
48.48.1.iy.1 $48$ $2$ $2$ $1$
48.48.1.jb.1 $48$ $2$ $2$ $1$
48.48.1.jd.1 $48$ $2$ $2$ $1$
48.48.1.je.1 $48$ $2$ $2$ $1$
48.48.1.jg.1 $48$ $2$ $2$ $1$
48.48.1.jj.1 $48$ $2$ $2$ $1$
48.48.3.gd.1 $48$ $2$ $2$ $3$
48.48.3.ge.1 $48$ $2$ $2$ $3$
48.48.3.gf.1 $48$ $2$ $2$ $3$
48.48.3.gg.1 $48$ $2$ $2$ $3$
48.72.0.d.1 $48$ $3$ $3$ $0$
48.96.7.cr.1 $48$ $4$ $4$ $7$
80.48.1.ix.1 $80$ $2$ $2$ $1$
80.48.1.iy.1 $80$ $2$ $2$ $1$
80.48.1.ja.1 $80$ $2$ $2$ $1$
80.48.1.jd.1 $80$ $2$ $2$ $1$
80.48.1.jf.1 $80$ $2$ $2$ $1$
80.48.1.jg.1 $80$ $2$ $2$ $1$
80.48.1.ji.1 $80$ $2$ $2$ $1$
80.48.1.jl.1 $80$ $2$ $2$ $1$
80.48.3.gx.2 $80$ $2$ $2$ $3$
80.48.3.gy.2 $80$ $2$ $2$ $3$
80.48.3.gz.2 $80$ $2$ $2$ $3$
80.48.3.ha.1 $80$ $2$ $2$ $3$
80.120.8.bn.1 $80$ $5$ $5$ $8$
80.144.7.df.1 $80$ $6$ $6$ $7$
80.240.15.dj.1 $80$ $10$ $10$ $15$
112.48.1.iv.1 $112$ $2$ $2$ $1$
112.48.1.iw.1 $112$ $2$ $2$ $1$
112.48.1.iy.1 $112$ $2$ $2$ $1$
112.48.1.jb.1 $112$ $2$ $2$ $1$
112.48.1.jd.1 $112$ $2$ $2$ $1$
112.48.1.je.1 $112$ $2$ $2$ $1$
112.48.1.jg.1 $112$ $2$ $2$ $1$
112.48.1.jj.1 $112$ $2$ $2$ $1$
112.48.3.gd.2 $112$ $2$ $2$ $3$
112.48.3.ge.2 $112$ $2$ $2$ $3$
112.48.3.gf.2 $112$ $2$ $2$ $3$
112.48.3.gg.1 $112$ $2$ $2$ $3$
112.192.15.bh.1 $112$ $8$ $8$ $15$
176.48.1.iv.1 $176$ $2$ $2$ $1$
176.48.1.iw.1 $176$ $2$ $2$ $1$
176.48.1.iy.1 $176$ $2$ $2$ $1$
176.48.1.jb.1 $176$ $2$ $2$ $1$
176.48.1.jd.1 $176$ $2$ $2$ $1$
176.48.1.je.1 $176$ $2$ $2$ $1$
176.48.1.jg.1 $176$ $2$ $2$ $1$
176.48.1.jj.1 $176$ $2$ $2$ $1$
176.48.3.gd.2 $176$ $2$ $2$ $3$
176.48.3.ge.2 $176$ $2$ $2$ $3$
176.48.3.gf.2 $176$ $2$ $2$ $3$
176.48.3.gg.1 $176$ $2$ $2$ $3$
176.288.23.bh.1 $176$ $12$ $12$ $23$
208.48.1.ix.1 $208$ $2$ $2$ $1$
208.48.1.iy.1 $208$ $2$ $2$ $1$
208.48.1.ja.1 $208$ $2$ $2$ $1$
208.48.1.jd.1 $208$ $2$ $2$ $1$
208.48.1.jf.1 $208$ $2$ $2$ $1$
208.48.1.jg.1 $208$ $2$ $2$ $1$
208.48.1.ji.1 $208$ $2$ $2$ $1$
208.48.1.jl.1 $208$ $2$ $2$ $1$
208.48.3.gp.2 $208$ $2$ $2$ $3$
208.48.3.gq.2 $208$ $2$ $2$ $3$
208.48.3.gr.2 $208$ $2$ $2$ $3$
208.48.3.gs.1 $208$ $2$ $2$ $3$
208.336.23.el.1 $208$ $14$ $14$ $23$
240.48.1.bbj.1 $240$ $2$ $2$ $1$
240.48.1.bbk.1 $240$ $2$ $2$ $1$
240.48.1.bbm.1 $240$ $2$ $2$ $1$
240.48.1.bbp.1 $240$ $2$ $2$ $1$
240.48.1.bbr.1 $240$ $2$ $2$ $1$
240.48.1.bbs.1 $240$ $2$ $2$ $1$
240.48.1.bbu.1 $240$ $2$ $2$ $1$
240.48.1.bbx.1 $240$ $2$ $2$ $1$
240.48.3.sx.1 $240$ $2$ $2$ $3$
240.48.3.sy.1 $240$ $2$ $2$ $3$
240.48.3.sz.1 $240$ $2$ $2$ $3$
240.48.3.ta.1 $240$ $2$ $2$ $3$
272.48.1.ix.1 $272$ $2$ $2$ $1$
272.48.1.iy.1 $272$ $2$ $2$ $1$
272.48.1.ja.1 $272$ $2$ $2$ $1$
272.48.1.jd.1 $272$ $2$ $2$ $1$
272.48.1.jf.1 $272$ $2$ $2$ $1$
272.48.1.jg.1 $272$ $2$ $2$ $1$
272.48.1.ji.1 $272$ $2$ $2$ $1$
272.48.1.jl.1 $272$ $2$ $2$ $1$
272.48.3.gp.1 $272$ $2$ $2$ $3$
272.48.3.gq.1 $272$ $2$ $2$ $3$
272.48.3.gr.2 $272$ $2$ $2$ $3$
272.48.3.gs.2 $272$ $2$ $2$ $3$
304.48.1.iv.1 $304$ $2$ $2$ $1$
304.48.1.iw.1 $304$ $2$ $2$ $1$
304.48.1.iy.1 $304$ $2$ $2$ $1$
304.48.1.jb.1 $304$ $2$ $2$ $1$
304.48.1.jd.1 $304$ $2$ $2$ $1$
304.48.1.je.1 $304$ $2$ $2$ $1$
304.48.1.jg.1 $304$ $2$ $2$ $1$
304.48.1.jj.1 $304$ $2$ $2$ $1$
304.48.3.gd.2 $304$ $2$ $2$ $3$
304.48.3.ge.2 $304$ $2$ $2$ $3$
304.48.3.gf.2 $304$ $2$ $2$ $3$
304.48.3.gg.1 $304$ $2$ $2$ $3$