Properties

Label 16.24.0-8.n.1.1
Level $16$
Index $24$
Genus $0$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $4$

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Invariants

Level: $16$ $\SL_2$-level: $16$
Index: $24$ $\PSL_2$-index:$12$
Genus: $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (all of which are rational) Cusp widths $1^{2}\cdot2\cdot8$ Cusp orbits $1^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8C0
Rouse and Zureick-Brown (RZB) label: X36g
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.24.0.10

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}1&4\\0&11\end{bmatrix}$, $\begin{bmatrix}7&4\\8&3\end{bmatrix}$, $\begin{bmatrix}9&11\\8&11\end{bmatrix}$, $\begin{bmatrix}11&6\\8&11\end{bmatrix}$, $\begin{bmatrix}13&2\\0&13\end{bmatrix}$
Contains $-I$: no $\quad$ (see 8.12.0.n.1 for the level structure with $-I$)
Cyclic 16-isogeny field degree: $2$
Cyclic 16-torsion field degree: $16$
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 5199 stored non-cuspidal points.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{x^{12}(x^{4}-16x^{2}y^{2}+16y^{4})^{3}}{y^{8}x^{14}(x-4y)(x+4y)}$

Modular covers

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

Covering curve Level Index Degree Genus
16.48.0-16.e.1.12 $16$ $2$ $2$ $0$
16.48.0-16.e.1.16 $16$ $2$ $2$ $0$
16.48.0-16.e.2.14 $16$ $2$ $2$ $0$
16.48.0-16.e.2.16 $16$ $2$ $2$ $0$
16.48.0-16.f.1.15 $16$ $2$ $2$ $0$
16.48.0-16.f.1.16 $16$ $2$ $2$ $0$
16.48.0-16.f.2.8 $16$ $2$ $2$ $0$
16.48.0-16.f.2.16 $16$ $2$ $2$ $0$
16.48.0-16.g.1.11 $16$ $2$ $2$ $0$
16.48.0-16.g.1.15 $16$ $2$ $2$ $0$
16.48.0-16.h.1.3 $16$ $2$ $2$ $0$
16.48.0-16.h.1.7 $16$ $2$ $2$ $0$
16.48.0-8.i.1.2 $16$ $2$ $2$ $0$
16.48.0-8.k.1.2 $16$ $2$ $2$ $0$
16.48.0-8.q.1.2 $16$ $2$ $2$ $0$
16.48.0-8.r.1.2 $16$ $2$ $2$ $0$
16.48.0-8.ba.1.1 $16$ $2$ $2$ $0$
16.48.0-8.ba.1.2 $16$ $2$ $2$ $0$
16.48.0-8.ba.2.1 $16$ $2$ $2$ $0$
16.48.0-8.ba.2.5 $16$ $2$ $2$ $0$
16.48.0-8.bb.1.1 $16$ $2$ $2$ $0$
16.48.0-8.bb.1.3 $16$ $2$ $2$ $0$
16.48.0-8.bb.2.1 $16$ $2$ $2$ $0$
16.48.0-8.bb.2.2 $16$ $2$ $2$ $0$
16.48.1-16.a.1.3 $16$ $2$ $2$ $1$
16.48.1-16.a.1.7 $16$ $2$ $2$ $1$
16.48.1-16.b.1.11 $16$ $2$ $2$ $1$
16.48.1-16.b.1.15 $16$ $2$ $2$ $1$
48.48.0-48.e.1.16 $48$ $2$ $2$ $0$
48.48.0-48.e.1.32 $48$ $2$ $2$ $0$
48.48.0-48.e.2.24 $48$ $2$ $2$ $0$
48.48.0-48.e.2.32 $48$ $2$ $2$ $0$
48.48.0-48.f.1.16 $48$ $2$ $2$ $0$
48.48.0-48.f.1.32 $48$ $2$ $2$ $0$
48.48.0-48.f.2.24 $48$ $2$ $2$ $0$
48.48.0-48.f.2.32 $48$ $2$ $2$ $0$
48.48.0-48.g.1.22 $48$ $2$ $2$ $0$
48.48.0-48.g.1.30 $48$ $2$ $2$ $0$
48.48.0-48.h.1.14 $48$ $2$ $2$ $0$
48.48.0-48.h.1.30 $48$ $2$ $2$ $0$
48.48.0-24.bh.1.7 $48$ $2$ $2$ $0$
48.48.0-24.bj.1.7 $48$ $2$ $2$ $0$
48.48.0-24.bl.1.8 $48$ $2$ $2$ $0$
48.48.0-24.bn.1.7 $48$ $2$ $2$ $0$
48.48.0-24.by.1.4 $48$ $2$ $2$ $0$
48.48.0-24.by.1.12 $48$ $2$ $2$ $0$
48.48.0-24.by.2.2 $48$ $2$ $2$ $0$
48.48.0-24.by.2.10 $48$ $2$ $2$ $0$
48.48.0-24.bz.1.2 $48$ $2$ $2$ $0$
48.48.0-24.bz.1.10 $48$ $2$ $2$ $0$
48.48.0-24.bz.2.4 $48$ $2$ $2$ $0$
48.48.0-24.bz.2.12 $48$ $2$ $2$ $0$
48.48.1-48.a.1.14 $48$ $2$ $2$ $1$
48.48.1-48.a.1.30 $48$ $2$ $2$ $1$
48.48.1-48.b.1.22 $48$ $2$ $2$ $1$
48.48.1-48.b.1.30 $48$ $2$ $2$ $1$
48.72.2-24.cj.1.11 $48$ $3$ $3$ $2$
48.96.1-24.ir.1.15 $48$ $4$ $4$ $1$
80.48.0-80.m.1.16 $80$ $2$ $2$ $0$
80.48.0-80.m.1.32 $80$ $2$ $2$ $0$
80.48.0-80.m.2.28 $80$ $2$ $2$ $0$
80.48.0-80.m.2.32 $80$ $2$ $2$ $0$
80.48.0-80.n.1.16 $80$ $2$ $2$ $0$
80.48.0-80.n.1.32 $80$ $2$ $2$ $0$
80.48.0-80.n.2.28 $80$ $2$ $2$ $0$
80.48.0-80.n.2.32 $80$ $2$ $2$ $0$
80.48.0-80.o.1.15 $80$ $2$ $2$ $0$
80.48.0-80.o.1.31 $80$ $2$ $2$ $0$
80.48.0-80.p.1.14 $80$ $2$ $2$ $0$
80.48.0-80.p.1.30 $80$ $2$ $2$ $0$
80.48.0-40.bj.1.4 $80$ $2$ $2$ $0$
80.48.0-40.bl.1.6 $80$ $2$ $2$ $0$
80.48.0-40.bn.1.4 $80$ $2$ $2$ $0$
80.48.0-40.bp.1.6 $80$ $2$ $2$ $0$
80.48.0-40.ca.1.1 $80$ $2$ $2$ $0$
80.48.0-40.ca.1.9 $80$ $2$ $2$ $0$
80.48.0-40.ca.2.1 $80$ $2$ $2$ $0$
80.48.0-40.ca.2.5 $80$ $2$ $2$ $0$
80.48.0-40.cb.1.1 $80$ $2$ $2$ $0$
80.48.0-40.cb.1.9 $80$ $2$ $2$ $0$
80.48.0-40.cb.2.1 $80$ $2$ $2$ $0$
80.48.0-40.cb.2.5 $80$ $2$ $2$ $0$
80.48.1-80.a.1.14 $80$ $2$ $2$ $1$
80.48.1-80.a.1.30 $80$ $2$ $2$ $1$
80.48.1-80.b.1.15 $80$ $2$ $2$ $1$
80.48.1-80.b.1.31 $80$ $2$ $2$ $1$
80.120.4-40.bl.1.12 $80$ $5$ $5$ $4$
80.144.3-40.bx.1.14 $80$ $6$ $6$ $3$
80.240.7-40.cj.1.32 $80$ $10$ $10$ $7$
112.48.0-112.e.1.23 $112$ $2$ $2$ $0$
112.48.0-112.e.1.31 $112$ $2$ $2$ $0$
112.48.0-112.e.2.23 $112$ $2$ $2$ $0$
112.48.0-112.e.2.31 $112$ $2$ $2$ $0$
112.48.0-112.f.1.23 $112$ $2$ $2$ $0$
112.48.0-112.f.1.31 $112$ $2$ $2$ $0$
112.48.0-112.f.2.27 $112$ $2$ $2$ $0$
112.48.0-112.f.2.31 $112$ $2$ $2$ $0$
112.48.0-112.g.1.24 $112$ $2$ $2$ $0$
112.48.0-112.g.1.32 $112$ $2$ $2$ $0$
112.48.0-112.h.1.24 $112$ $2$ $2$ $0$
112.48.0-112.h.1.32 $112$ $2$ $2$ $0$
112.48.0-56.bf.1.3 $112$ $2$ $2$ $0$
112.48.0-56.bh.1.8 $112$ $2$ $2$ $0$
112.48.0-56.bj.1.3 $112$ $2$ $2$ $0$
112.48.0-56.bl.1.8 $112$ $2$ $2$ $0$
112.48.0-56.bu.1.8 $112$ $2$ $2$ $0$
112.48.0-56.bu.1.16 $112$ $2$ $2$ $0$
112.48.0-56.bu.2.12 $112$ $2$ $2$ $0$
112.48.0-56.bu.2.16 $112$ $2$ $2$ $0$
112.48.0-56.bv.1.8 $112$ $2$ $2$ $0$
112.48.0-56.bv.1.16 $112$ $2$ $2$ $0$
112.48.0-56.bv.2.14 $112$ $2$ $2$ $0$
112.48.0-56.bv.2.16 $112$ $2$ $2$ $0$
112.48.1-112.a.1.24 $112$ $2$ $2$ $1$
112.48.1-112.a.1.32 $112$ $2$ $2$ $1$
112.48.1-112.b.1.24 $112$ $2$ $2$ $1$
112.48.1-112.b.1.32 $112$ $2$ $2$ $1$
112.192.5-56.bl.1.5 $112$ $8$ $8$ $5$
112.504.16-56.cj.1.32 $112$ $21$ $21$ $16$
176.48.0-176.e.1.24 $176$ $2$ $2$ $0$
176.48.0-176.e.1.32 $176$ $2$ $2$ $0$
176.48.0-176.e.2.24 $176$ $2$ $2$ $0$
176.48.0-176.e.2.32 $176$ $2$ $2$ $0$
176.48.0-176.f.1.24 $176$ $2$ $2$ $0$
176.48.0-176.f.1.32 $176$ $2$ $2$ $0$
176.48.0-176.f.2.28 $176$ $2$ $2$ $0$
176.48.0-176.f.2.32 $176$ $2$ $2$ $0$
176.48.0-176.g.1.27 $176$ $2$ $2$ $0$
176.48.0-176.g.1.31 $176$ $2$ $2$ $0$
176.48.0-176.h.1.23 $176$ $2$ $2$ $0$
176.48.0-176.h.1.31 $176$ $2$ $2$ $0$
176.48.0-88.bf.1.3 $176$ $2$ $2$ $0$
176.48.0-88.bh.1.8 $176$ $2$ $2$ $0$
176.48.0-88.bj.1.3 $176$ $2$ $2$ $0$
176.48.0-88.bl.1.8 $176$ $2$ $2$ $0$
176.48.0-88.bu.1.2 $176$ $2$ $2$ $0$
176.48.0-88.bu.1.6 $176$ $2$ $2$ $0$
176.48.0-88.bu.2.3 $176$ $2$ $2$ $0$
176.48.0-88.bu.2.7 $176$ $2$ $2$ $0$
176.48.0-88.bv.1.3 $176$ $2$ $2$ $0$
176.48.0-88.bv.1.7 $176$ $2$ $2$ $0$
176.48.0-88.bv.2.2 $176$ $2$ $2$ $0$
176.48.0-88.bv.2.4 $176$ $2$ $2$ $0$
176.48.1-176.a.1.23 $176$ $2$ $2$ $1$
176.48.1-176.a.1.31 $176$ $2$ $2$ $1$
176.48.1-176.b.1.27 $176$ $2$ $2$ $1$
176.48.1-176.b.1.31 $176$ $2$ $2$ $1$
176.288.9-88.bl.1.1 $176$ $12$ $12$ $9$
208.48.0-208.m.1.23 $208$ $2$ $2$ $0$
208.48.0-208.m.1.31 $208$ $2$ $2$ $0$
208.48.0-208.m.2.23 $208$ $2$ $2$ $0$
208.48.0-208.m.2.31 $208$ $2$ $2$ $0$
208.48.0-208.n.1.23 $208$ $2$ $2$ $0$
208.48.0-208.n.1.31 $208$ $2$ $2$ $0$
208.48.0-208.n.2.27 $208$ $2$ $2$ $0$
208.48.0-208.n.2.31 $208$ $2$ $2$ $0$
208.48.0-208.o.1.28 $208$ $2$ $2$ $0$
208.48.0-208.o.1.32 $208$ $2$ $2$ $0$
208.48.0-208.p.1.24 $208$ $2$ $2$ $0$
208.48.0-208.p.1.32 $208$ $2$ $2$ $0$
208.48.0-104.bj.1.4 $208$ $2$ $2$ $0$
208.48.0-104.bl.1.6 $208$ $2$ $2$ $0$
208.48.0-104.bn.1.4 $208$ $2$ $2$ $0$
208.48.0-104.bp.1.6 $208$ $2$ $2$ $0$
208.48.0-104.ca.1.1 $208$ $2$ $2$ $0$
208.48.0-104.ca.1.5 $208$ $2$ $2$ $0$
208.48.0-104.ca.2.5 $208$ $2$ $2$ $0$
208.48.0-104.ca.2.7 $208$ $2$ $2$ $0$
208.48.0-104.cb.1.3 $208$ $2$ $2$ $0$
208.48.0-104.cb.1.7 $208$ $2$ $2$ $0$
208.48.0-104.cb.2.1 $208$ $2$ $2$ $0$
208.48.0-104.cb.2.5 $208$ $2$ $2$ $0$
208.48.1-208.a.1.24 $208$ $2$ $2$ $1$
208.48.1-208.a.1.32 $208$ $2$ $2$ $1$
208.48.1-208.b.1.28 $208$ $2$ $2$ $1$
208.48.1-208.b.1.32 $208$ $2$ $2$ $1$
208.336.11-104.bx.1.17 $208$ $14$ $14$ $11$
240.48.0-240.m.1.31 $240$ $2$ $2$ $0$
240.48.0-240.m.1.63 $240$ $2$ $2$ $0$
240.48.0-240.m.2.55 $240$ $2$ $2$ $0$
240.48.0-240.m.2.63 $240$ $2$ $2$ $0$
240.48.0-240.n.1.31 $240$ $2$ $2$ $0$
240.48.0-240.n.1.63 $240$ $2$ $2$ $0$
240.48.0-240.n.2.47 $240$ $2$ $2$ $0$
240.48.0-240.n.2.63 $240$ $2$ $2$ $0$
240.48.0-240.o.1.32 $240$ $2$ $2$ $0$
240.48.0-240.o.1.64 $240$ $2$ $2$ $0$
240.48.0-240.p.1.32 $240$ $2$ $2$ $0$
240.48.0-240.p.1.64 $240$ $2$ $2$ $0$
240.48.0-120.dd.1.14 $240$ $2$ $2$ $0$
240.48.0-120.df.1.10 $240$ $2$ $2$ $0$
240.48.0-120.dh.1.13 $240$ $2$ $2$ $0$
240.48.0-120.dj.1.10 $240$ $2$ $2$ $0$
240.48.0-120.ei.1.15 $240$ $2$ $2$ $0$
240.48.0-120.ei.1.31 $240$ $2$ $2$ $0$
240.48.0-120.ei.2.27 $240$ $2$ $2$ $0$
240.48.0-120.ei.2.31 $240$ $2$ $2$ $0$
240.48.0-120.ej.1.15 $240$ $2$ $2$ $0$
240.48.0-120.ej.1.31 $240$ $2$ $2$ $0$
240.48.0-120.ej.2.27 $240$ $2$ $2$ $0$
240.48.0-120.ej.2.31 $240$ $2$ $2$ $0$
240.48.1-240.a.1.32 $240$ $2$ $2$ $1$
240.48.1-240.a.1.64 $240$ $2$ $2$ $1$
240.48.1-240.b.1.32 $240$ $2$ $2$ $1$
240.48.1-240.b.1.64 $240$ $2$ $2$ $1$
272.48.0-272.m.1.15 $272$ $2$ $2$ $0$
272.48.0-272.m.1.31 $272$ $2$ $2$ $0$
272.48.0-272.m.2.15 $272$ $2$ $2$ $0$
272.48.0-272.m.2.31 $272$ $2$ $2$ $0$
272.48.0-272.n.1.15 $272$ $2$ $2$ $0$
272.48.0-272.n.1.31 $272$ $2$ $2$ $0$
272.48.0-272.n.2.23 $272$ $2$ $2$ $0$
272.48.0-272.n.2.31 $272$ $2$ $2$ $0$
272.48.0-272.o.1.7 $272$ $2$ $2$ $0$
272.48.0-272.o.1.15 $272$ $2$ $2$ $0$
272.48.0-272.p.1.11 $272$ $2$ $2$ $0$
272.48.0-272.p.1.15 $272$ $2$ $2$ $0$
272.48.0-136.bj.1.4 $272$ $2$ $2$ $0$
272.48.0-136.bl.1.4 $272$ $2$ $2$ $0$
272.48.0-136.bn.1.4 $272$ $2$ $2$ $0$
272.48.0-136.bp.1.4 $272$ $2$ $2$ $0$
272.48.0-136.ca.1.4 $272$ $2$ $2$ $0$
272.48.0-136.ca.1.12 $272$ $2$ $2$ $0$
272.48.0-136.ca.2.10 $272$ $2$ $2$ $0$
272.48.0-136.ca.2.14 $272$ $2$ $2$ $0$
272.48.0-136.cb.1.6 $272$ $2$ $2$ $0$
272.48.0-136.cb.1.14 $272$ $2$ $2$ $0$
272.48.0-136.cb.2.6 $272$ $2$ $2$ $0$
272.48.0-136.cb.2.8 $272$ $2$ $2$ $0$
272.48.1-272.a.1.11 $272$ $2$ $2$ $1$
272.48.1-272.a.1.15 $272$ $2$ $2$ $1$
272.48.1-272.b.1.7 $272$ $2$ $2$ $1$
272.48.1-272.b.1.15 $272$ $2$ $2$ $1$
272.432.15-136.cj.1.16 $272$ $18$ $18$ $15$
304.48.0-304.e.1.24 $304$ $2$ $2$ $0$
304.48.0-304.e.1.32 $304$ $2$ $2$ $0$
304.48.0-304.e.2.24 $304$ $2$ $2$ $0$
304.48.0-304.e.2.32 $304$ $2$ $2$ $0$
304.48.0-304.f.1.24 $304$ $2$ $2$ $0$
304.48.0-304.f.1.32 $304$ $2$ $2$ $0$
304.48.0-304.f.2.28 $304$ $2$ $2$ $0$
304.48.0-304.f.2.32 $304$ $2$ $2$ $0$
304.48.0-304.g.1.27 $304$ $2$ $2$ $0$
304.48.0-304.g.1.31 $304$ $2$ $2$ $0$
304.48.0-304.h.1.23 $304$ $2$ $2$ $0$
304.48.0-304.h.1.31 $304$ $2$ $2$ $0$
304.48.0-152.bf.1.5 $304$ $2$ $2$ $0$
304.48.0-152.bh.1.7 $304$ $2$ $2$ $0$
304.48.0-152.bj.1.4 $304$ $2$ $2$ $0$
304.48.0-152.bl.1.7 $304$ $2$ $2$ $0$
304.48.0-152.bu.1.12 $304$ $2$ $2$ $0$
304.48.0-152.bu.1.16 $304$ $2$ $2$ $0$
304.48.0-152.bu.2.11 $304$ $2$ $2$ $0$
304.48.0-152.bu.2.15 $304$ $2$ $2$ $0$
304.48.0-152.bv.1.11 $304$ $2$ $2$ $0$
304.48.0-152.bv.1.15 $304$ $2$ $2$ $0$
304.48.0-152.bv.2.14 $304$ $2$ $2$ $0$
304.48.0-152.bv.2.16 $304$ $2$ $2$ $0$
304.48.1-304.a.1.23 $304$ $2$ $2$ $1$
304.48.1-304.a.1.31 $304$ $2$ $2$ $1$
304.48.1-304.b.1.27 $304$ $2$ $2$ $1$
304.48.1-304.b.1.31 $304$ $2$ $2$ $1$
304.480.17-152.bl.1.3 $304$ $20$ $20$ $17$