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: | 48.144.4.2438 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&32\\32&13\end{bmatrix}$, $\begin{bmatrix}23&34\\40&7\end{bmatrix}$, $\begin{bmatrix}25&18\\24&23\end{bmatrix}$, $\begin{bmatrix}31&32\\8&35\end{bmatrix}$, $\begin{bmatrix}47&21\\0&17\end{bmatrix}$, $\begin{bmatrix}47&24\\0&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.4.fd.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 $ | $=$ | $ 15 y^{2} - 2 y w + 3 z^{2} - w^{2} $ |
| $=$ | $3 x^{3} + y^{2} z + y z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 385 x^{6} + 27 x^{5} z - 285 x^{4} z^{2} - 48 x^{3} y^{3} - 80 x^{3} z^{3} - 72 x^{2} y^{3} z + \cdots + z^{6} $ |
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 |
|---|
| $(0:1/3:0:1)$, $(0:-1/5:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^3\cdot5^2}\cdot\frac{230291100000yz^{10}w-259659448200000yz^{8}w^{3}+1427759167104000yz^{6}w^{5}+194177755699200yz^{4}w^{7}+8557798809600yz^{2}w^{9}+124727902208yw^{11}+553584375z^{12}-6273956250000z^{10}w^{2}-464042318310000z^{8}w^{4}+260152515417600z^{6}w^{6}+37614580727040z^{4}w^{8}+1694173360128z^{2}w^{10}+25090699264w^{12}}{z^{2}(164025000yz^{8}w-506655000yz^{6}w^{3}+449695800yz^{4}w^{5}-157807080yz^{2}w^{7}+19413152yw^{9}-12301875z^{10}+158557500z^{8}w^{2}-246584250z^{6}w^{4}+148703580z^{4}w^{6}-39727083z^{2}w^{8}+3945616w^{10})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.72.4.fd.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y+\frac{1}{9}w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z+\frac{4}{9}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 385X^{6}-48X^{3}Y^{3}+27X^{5}Z-72X^{2}Y^{3}Z-285X^{4}Z^{2}-36XY^{3}Z^{2}-80X^{3}Z^{3}-6Y^{3}Z^{3}-45X^{2}Z^{4}-3XZ^{5}+Z^{6} $ |
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.bl.1.3 | $48$ | $3$ | $3$ | $0$ | $0$ | 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.vh.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.7-24.vj.1.2 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-24.vp.1.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.7-24.vr.1.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.7-24.yj.1.7 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-24.yl.1.13 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.7-24.yr.1.7 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-24.yt.1.7 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 48.288.8-48.dk.1.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dk.1.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dk.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dk.2.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dl.1.7 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dl.1.23 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dl.2.7 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dl.2.15 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dm.1.7 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dm.1.23 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dm.2.7 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dm.2.15 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dn.1.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dn.1.9 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dn.2.1 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-48.dn.2.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.go.1.3 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.go.1.7 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.go.2.6 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.go.2.8 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gp.1.7 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gp.1.8 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gp.2.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gp.2.6 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gq.1.7 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gq.1.8 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gq.2.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gq.2.6 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gr.1.4 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gr.1.8 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gr.2.5 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.8-24.gr.2.7 | $48$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
| 48.288.9-48.cq.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.cq.1.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.cs.1.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.cs.1.21 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.gy.1.10 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.gy.1.26 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.gz.1.10 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 48.288.9-48.gz.1.26 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 48.288.9-48.ka.1.13 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.ka.1.29 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.kb.1.11 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.kb.1.27 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.lw.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.lw.1.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.ly.1.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.ly.1.9 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 144.432.16-72.fd.1.12 | $144$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 240.288.7-120.dga.1.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dgc.1.6 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dgi.1.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dgk.1.6 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dmy.1.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dna.1.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dng.1.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dni.1.13 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.8-240.ei.1.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ei.1.17 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ei.2.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ei.2.9 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ej.1.10 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ej.1.42 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ej.2.10 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ej.2.26 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ek.1.10 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ek.1.42 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ek.2.10 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ek.2.26 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.el.1.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.el.1.17 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.el.2.1 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.el.2.9 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qs.1.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qs.1.12 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qs.2.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qs.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qt.1.23 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qt.1.24 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qt.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qt.2.8 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qu.1.23 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qu.1.24 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qu.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qu.2.8 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qv.1.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qv.1.11 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qv.2.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qv.2.8 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.9-240.ie.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ie.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.if.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.if.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.jk.1.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.jk.1.42 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.jl.1.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.jl.1.38 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mq.1.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mq.1.42 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mr.1.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mr.1.38 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.om.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.om.1.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.on.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.on.1.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |