↑

Properties

Label 48.24.0-8.n.1.1
Level $48$
Index $24$
Genus $0$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $4$

Related objects

Downloads

Learn more

Invariants

Level: $48$ $\SL_2$-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, Sutherland, and Zureick-Brown (RSZB) label: 48.24.0.11

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}3&46\\40&27\end{bmatrix}$, $\begin{bmatrix}5&12\\40&25\end{bmatrix}$, $\begin{bmatrix}15&5\\32&17\end{bmatrix}$, $\begin{bmatrix}15&25\\8&33\end{bmatrix}$, $\begin{bmatrix}31&23\\32&17\end{bmatrix}$, $\begin{bmatrix}37&0\\36&23\end{bmatrix}$
Contains $-I$: no $\quad$ (see 8.12.0.n.1 for the level structure with $-I$)
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $49152$

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
48.48.0-16.e.1.9 $48$ $2$ $2$ $0$
48.48.0-16.e.1.10 $48$ $2$ $2$ $0$
48.48.0-16.e.2.2 $48$ $2$ $2$ $0$
48.48.0-16.e.2.10 $48$ $2$ $2$ $0$
48.48.0-48.e.1.13 $48$ $2$ $2$ $0$
48.48.0-48.e.1.29 $48$ $2$ $2$ $0$
48.48.0-48.e.2.21 $48$ $2$ $2$ $0$
48.48.0-48.e.2.29 $48$ $2$ $2$ $0$
48.48.0-16.f.1.9 $48$ $2$ $2$ $0$
48.48.0-16.f.1.13 $48$ $2$ $2$ $0$
48.48.0-16.f.2.3 $48$ $2$ $2$ $0$
48.48.0-16.f.2.4 $48$ $2$ $2$ $0$
48.48.0-48.f.1.13 $48$ $2$ $2$ $0$
48.48.0-48.f.1.29 $48$ $2$ $2$ $0$
48.48.0-48.f.2.21 $48$ $2$ $2$ $0$
48.48.0-48.f.2.29 $48$ $2$ $2$ $0$
48.48.0-16.g.1.5 $48$ $2$ $2$ $0$
48.48.0-16.g.1.6 $48$ $2$ $2$ $0$
48.48.0-48.g.1.23 $48$ $2$ $2$ $0$
48.48.0-48.g.1.31 $48$ $2$ $2$ $0$
48.48.0-16.h.1.13 $48$ $2$ $2$ $0$
48.48.0-16.h.1.14 $48$ $2$ $2$ $0$
48.48.0-48.h.1.15 $48$ $2$ $2$ $0$
48.48.0-48.h.1.31 $48$ $2$ $2$ $0$
48.48.0-8.i.1.4 $48$ $2$ $2$ $0$
48.48.0-8.k.1.3 $48$ $2$ $2$ $0$
48.48.0-8.q.1.4 $48$ $2$ $2$ $0$
48.48.0-8.r.1.3 $48$ $2$ $2$ $0$
48.48.0-8.ba.1.2 $48$ $2$ $2$ $0$
48.48.0-8.ba.1.6 $48$ $2$ $2$ $0$
48.48.0-8.ba.2.2 $48$ $2$ $2$ $0$
48.48.0-8.ba.2.4 $48$ $2$ $2$ $0$
48.48.0-8.bb.1.2 $48$ $2$ $2$ $0$
48.48.0-8.bb.1.4 $48$ $2$ $2$ $0$
48.48.0-8.bb.2.2 $48$ $2$ $2$ $0$
48.48.0-8.bb.2.4 $48$ $2$ $2$ $0$
48.48.0-24.bh.1.8 $48$ $2$ $2$ $0$
48.48.0-24.bj.1.7 $48$ $2$ $2$ $0$
48.48.0-24.bl.1.7 $48$ $2$ $2$ $0$
48.48.0-24.bn.1.7 $48$ $2$ $2$ $0$
48.48.0-24.by.1.3 $48$ $2$ $2$ $0$
48.48.0-24.by.1.11 $48$ $2$ $2$ $0$
48.48.0-24.by.2.1 $48$ $2$ $2$ $0$
48.48.0-24.by.2.9 $48$ $2$ $2$ $0$
48.48.0-24.bz.1.1 $48$ $2$ $2$ $0$
48.48.0-24.bz.1.9 $48$ $2$ $2$ $0$
48.48.0-24.bz.2.3 $48$ $2$ $2$ $0$
48.48.0-24.bz.2.11 $48$ $2$ $2$ $0$
48.48.1-16.a.1.13 $48$ $2$ $2$ $1$
48.48.1-16.a.1.14 $48$ $2$ $2$ $1$
48.48.1-48.a.1.15 $48$ $2$ $2$ $1$
48.48.1-48.a.1.31 $48$ $2$ $2$ $1$
48.48.1-16.b.1.5 $48$ $2$ $2$ $1$
48.48.1-16.b.1.6 $48$ $2$ $2$ $1$
48.48.1-48.b.1.23 $48$ $2$ $2$ $1$
48.48.1-48.b.1.31 $48$ $2$ $2$ $1$
48.72.2-24.cj.1.14 $48$ $3$ $3$ $2$
48.96.1-24.ir.1.4 $48$ $4$ $4$ $1$
240.48.0-80.m.1.13 $240$ $2$ $2$ $0$
240.48.0-80.m.1.15 $240$ $2$ $2$ $0$
240.48.0-80.m.2.13 $240$ $2$ $2$ $0$
240.48.0-80.m.2.14 $240$ $2$ $2$ $0$
240.48.0-240.m.1.12 $240$ $2$ $2$ $0$
240.48.0-240.m.1.44 $240$ $2$ $2$ $0$
240.48.0-240.m.2.20 $240$ $2$ $2$ $0$
240.48.0-240.m.2.28 $240$ $2$ $2$ $0$
240.48.0-80.n.1.13 $240$ $2$ $2$ $0$
240.48.0-80.n.1.14 $240$ $2$ $2$ $0$
240.48.0-80.n.2.13 $240$ $2$ $2$ $0$
240.48.0-80.n.2.14 $240$ $2$ $2$ $0$
240.48.0-240.n.1.12 $240$ $2$ $2$ $0$
240.48.0-240.n.1.44 $240$ $2$ $2$ $0$
240.48.0-240.n.2.12 $240$ $2$ $2$ $0$
240.48.0-240.n.2.28 $240$ $2$ $2$ $0$
240.48.0-80.o.1.2 $240$ $2$ $2$ $0$
240.48.0-80.o.1.4 $240$ $2$ $2$ $0$
240.48.0-240.o.1.23 $240$ $2$ $2$ $0$
240.48.0-240.o.1.55 $240$ $2$ $2$ $0$
240.48.0-80.p.1.2 $240$ $2$ $2$ $0$
240.48.0-80.p.1.6 $240$ $2$ $2$ $0$
240.48.0-240.p.1.15 $240$ $2$ $2$ $0$
240.48.0-240.p.1.47 $240$ $2$ $2$ $0$
240.48.0-40.bj.1.7 $240$ $2$ $2$ $0$
240.48.0-40.bl.1.6 $240$ $2$ $2$ $0$
240.48.0-40.bn.1.7 $240$ $2$ $2$ $0$
240.48.0-40.bp.1.6 $240$ $2$ $2$ $0$
240.48.0-40.ca.1.11 $240$ $2$ $2$ $0$
240.48.0-40.ca.1.12 $240$ $2$ $2$ $0$
240.48.0-40.ca.2.10 $240$ $2$ $2$ $0$
240.48.0-40.ca.2.12 $240$ $2$ $2$ $0$
240.48.0-40.cb.1.10 $240$ $2$ $2$ $0$
240.48.0-40.cb.1.12 $240$ $2$ $2$ $0$
240.48.0-40.cb.2.11 $240$ $2$ $2$ $0$
240.48.0-40.cb.2.12 $240$ $2$ $2$ $0$
240.48.0-120.dd.1.12 $240$ $2$ $2$ $0$
240.48.0-120.df.1.12 $240$ $2$ $2$ $0$
240.48.0-120.dh.1.16 $240$ $2$ $2$ $0$
240.48.0-120.dj.1.12 $240$ $2$ $2$ $0$
240.48.0-120.ei.1.5 $240$ $2$ $2$ $0$
240.48.0-120.ei.1.21 $240$ $2$ $2$ $0$
240.48.0-120.ei.2.9 $240$ $2$ $2$ $0$
240.48.0-120.ei.2.13 $240$ $2$ $2$ $0$
240.48.0-120.ej.1.5 $240$ $2$ $2$ $0$
240.48.0-120.ej.1.21 $240$ $2$ $2$ $0$
240.48.0-120.ej.2.9 $240$ $2$ $2$ $0$
240.48.0-120.ej.2.13 $240$ $2$ $2$ $0$
240.48.1-80.a.1.2 $240$ $2$ $2$ $1$
240.48.1-80.a.1.6 $240$ $2$ $2$ $1$
240.48.1-240.a.1.11 $240$ $2$ $2$ $1$
240.48.1-240.a.1.43 $240$ $2$ $2$ $1$
240.48.1-80.b.1.2 $240$ $2$ $2$ $1$
240.48.1-80.b.1.4 $240$ $2$ $2$ $1$
240.48.1-240.b.1.19 $240$ $2$ $2$ $1$
240.48.1-240.b.1.51 $240$ $2$ $2$ $1$
240.120.4-40.bl.1.15 $240$ $5$ $5$ $4$
240.144.3-40.bx.1.30 $240$ $6$ $6$ $3$
240.240.7-40.cj.1.19 $240$ $10$ $10$ $7$