Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $288$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $5 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $6^{2}\cdot12\cdot48$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48A5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.144.5.7 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&8\\40&37\end{bmatrix}$, $\begin{bmatrix}1&17\\28&1\end{bmatrix}$, $\begin{bmatrix}1&28\\40&5\end{bmatrix}$, $\begin{bmatrix}5&11\\8&19\end{bmatrix}$, $\begin{bmatrix}13&41\\8&47\end{bmatrix}$, $\begin{bmatrix}37&10\\8&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.72.5.b.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $8192$ |
Jacobian
| Conductor: | $2^{19}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}$ |
| Newforms: | 32.2.a.a, 36.2.a.a$^{2}$, 288.2.a.a, 288.2.a.e |
Models
Embedded model Embedded model in $\mathbb{P}^{6}$
| $ 0 $ | $=$ | $ - y v^{2} + u^{2} v $ |
| $=$ | $ - x v^{2} + u^{3}$ | |
| $=$ | $x u v + t v^{2}$ | |
| $=$ | $x y v + t u v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 8 x^{11} - 2 x y^{2} z^{8} + y z^{10} $ |
Weierstrass model Weierstrass model
| $ y^{2} + x^{6} y $ | $=$ | $ 16 $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:0:0:0:1)$, $(0:0:0:1:0:0:0)$, $(0:0:-1:1:0:0:0)$, $(0:0:1:1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{12zw^{4}tv+13zwuv^{4}+76zt^{3}v^{3}-4zv^{6}-w^{7}-48w^{3}t^{2}v^{2}}{vtw^{4}z}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.72.5.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle u$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle v$ |
Equation of the image curve:
| $0$ | $=$ | $ 8X^{11}-2XY^{2}Z^{8}+YZ^{10} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 48.72.5.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle v$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4wuv^{4}-v^{6}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -u$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 16.48.1-16.b.1.2 | $16$ | $3$ | $3$ | $1$ | $0$ | $1^{4}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.1-16.b.1.2 | $16$ | $3$ | $3$ | $1$ | $0$ | $1^{4}$ |
| 24.72.2-24.cj.1.42 | $24$ | $2$ | $2$ | $2$ | $0$ | $1^{3}$ |
| 48.72.2-24.cj.1.30 | $48$ | $2$ | $2$ | $2$ | $0$ | $1^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.288.9-48.h.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.x.1.10 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 48.288.9-48.bf.1.41 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.bn.1.13 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 48.288.9-48.co.1.13 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.cp.1.13 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 48.288.9-48.cs.1.21 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.ct.1.21 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 48.288.9-48.de.1.2 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.288.9-48.df.1.1 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 48.288.9-48.di.1.2 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.288.9-48.dj.1.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.288.9-48.du.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.dv.1.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.288.9-48.dy.1.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.dz.1.3 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.288.9-48.ey.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.ey.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.ez.1.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.ez.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fa.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fa.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fb.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fb.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fc.1.13 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fc.2.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fd.1.10 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fd.2.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fe.1.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fe.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.ff.1.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.ff.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fg.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fg.2.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fh.1.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fh.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fi.1.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fi.2.10 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fj.1.13 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fj.2.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fk.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fk.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fl.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fl.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fm.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fm.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fn.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fn.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.288.9-48.fq.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.fr.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.fu.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.fv.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.gg.1.3 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 48.288.9-48.gh.1.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.288.9-48.gk.1.1 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 48.288.9-48.gl.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 96.288.11-96.a.1.62 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.a.2.62 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.c.1.62 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.c.2.62 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.e.1.2 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.e.2.2 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.g.1.2 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.g.2.2 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.i.1.3 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.i.2.3 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.j.1.1 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.j.2.1 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.k.1.13 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.k.2.9 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.l.1.13 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.l.2.9 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.m.1.15 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.m.2.13 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.n.1.9 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.n.2.1 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.o.1.3 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.o.2.3 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.p.1.1 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.p.2.1 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.q.1.2 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.q.2.4 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.s.1.2 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.s.2.4 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.u.1.1 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.u.2.1 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.w.1.1 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 96.288.11-96.w.2.1 | $96$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 144.432.17-144.b.1.1 | $144$ | $3$ | $3$ | $17$ | $?$ | not computed |
| 240.288.9-240.cw.1.34 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.cx.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.da.1.49 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.db.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.dm.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.dn.1.49 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.dq.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.dr.1.49 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ec.1.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ed.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.eg.1.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.eh.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.es.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.et.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ew.1.12 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ex.1.12 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.fw.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.fw.2.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.fx.1.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.fx.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.fy.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.fy.2.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.fz.1.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.fz.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ga.1.36 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ga.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gb.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gb.2.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gc.1.36 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gc.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gd.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gd.2.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ge.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ge.2.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gf.1.36 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gf.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gg.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gg.2.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gh.1.36 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gh.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gi.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gi.2.19 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gj.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gj.2.19 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gk.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gk.2.19 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gl.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gl.2.19 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.go.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gp.1.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gs.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.gt.1.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.he.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.hf.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.hi.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.hj.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |