Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $144$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $6^{4}\cdot24^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24D4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.144.4.2414 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&42\\0&13\end{bmatrix}$, $\begin{bmatrix}29&2\\32&25\end{bmatrix}$, $\begin{bmatrix}29&33\\0&1\end{bmatrix}$, $\begin{bmatrix}31&6\\24&19\end{bmatrix}$, $\begin{bmatrix}43&7\\8&31\end{bmatrix}$, $\begin{bmatrix}47&9\\24&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.4.ez.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: | $2^{10}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 36.2.a.a$^{3}$, 144.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 4 y^{2} - z^{2} - z w - w^{2} $ |
| $=$ | $3 x^{3} + 2 y z w + y w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{4} z + 4 x^{3} z^{2} + 24 x^{2} y^{3} - 3 x^{2} z^{3} - 24 x y^{3} z + x z^{4} + 6 y^{3} z^{2} + z^{5} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:1/2:1:0)$, $(0:-1/2:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^8}{3}\cdot\frac{(z^{2}-2zw-2w^{2})^{3}(z^{2}+4zw+w^{2})^{3}}{w^{2}(2z+w)^{2}(z^{2}+zw+w^{2})^{4}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.72.4.ez.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y-\frac{1}{2}z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 24X^{2}Y^{3}-2X^{4}Z-24XY^{3}Z+4X^{3}Z^{2}+6Y^{3}Z^{2}-3X^{2}Z^{3}+XZ^{4}+Z^{5} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.48.0-24.bh.1.1 | $48$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 48.72.2-24.cj.1.25 | $48$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 48.72.2-24.cj.1.30 | $48$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.288.7-24.ur.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.7-24.ut.1.2 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-24.uz.1.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.7-24.vb.1.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.7-24.xt.1.5 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-24.xv.1.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.7-24.yb.1.5 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-24.yd.1.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.8-48.dc.1.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dc.1.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dc.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dc.2.2 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dd.1.7 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dd.1.15 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dd.2.13 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dd.2.14 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.de.1.7 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.de.1.15 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.de.2.13 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.de.2.15 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.df.1.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.df.1.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.df.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.df.2.3 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gg.1.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gg.1.7 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gg.2.6 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gg.2.8 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gh.1.7 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gh.1.8 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gh.2.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gh.2.6 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gi.1.7 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gi.1.8 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gi.2.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gi.2.6 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gj.1.6 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gj.1.8 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gj.2.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gj.2.7 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.9-48.cm.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.cm.1.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.co.1.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.co.1.13 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.gu.1.18 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.gu.1.26 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.gv.1.18 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 48.288.9-48.gv.1.26 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 48.288.9-48.jw.1.21 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.jw.1.29 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.jx.1.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.jx.1.27 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.ls.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.ls.1.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.lu.1.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.lu.1.9 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 144.432.16-72.ez.1.10 | $144$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 240.288.7-120.dfk.1.6 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dfm.1.6 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dfs.1.6 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dfu.1.6 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dmi.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dmk.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dmq.1.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dms.1.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.8-240.ea.1.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ea.1.33 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ea.2.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ea.2.5 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.eb.1.10 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.eb.1.26 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.eb.2.18 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.eb.2.22 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ec.1.10 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ec.1.26 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ec.2.18 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ec.2.22 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ed.1.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ed.1.33 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ed.2.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ed.2.5 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qk.1.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qk.1.12 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qk.2.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qk.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.ql.1.23 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.ql.1.24 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.ql.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.ql.2.8 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qm.1.23 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qm.1.24 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qm.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qm.2.8 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qn.1.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qn.1.11 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qn.2.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qn.2.8 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.9-240.ia.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ia.1.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ib.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ib.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.jg.1.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.jg.1.42 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.jh.1.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.jh.1.38 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mm.1.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mm.1.42 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mn.1.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mn.1.38 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.oi.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.oi.1.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.oj.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.oj.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |