Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $576$ | ||
| 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: | $1$ | ||||||
| $\Q$-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: | 48.144.4.2642 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}11&17\\8&31\end{bmatrix}$, $\begin{bmatrix}21&34\\40&3\end{bmatrix}$, $\begin{bmatrix}41&8\\40&47\end{bmatrix}$, $\begin{bmatrix}41&19\\16&37\end{bmatrix}$, $\begin{bmatrix}45&2\\40&39\end{bmatrix}$, $\begin{bmatrix}45&40\\16&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.4.fb.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: | $2^{16}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 36.2.a.a$^{2}$, 576.2.a.e, 576.2.a.f |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 16 y^{2} + 6 z^{2} - w^{2} $ |
| $=$ | $6 x^{3} - y z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{4} z + 12 x^{2} y^{3} + 36 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 |
|---|
| $(0:-1/4:0:1)$, $(0:1/4:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^3}{3}\cdot\frac{(36z^{4}+84z^{2}w^{2}+w^{4})^{3}}{w^{2}z^{2}(6z^{2}-w^{2})^{4}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.72.4.fb.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y+\frac{1}{4}w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 12X^{2}Y^{3}-X^{4}Z+36Z^{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.bj.1.6 | $48$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 48.72.2-24.cj.1.27 | $48$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 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.uv.1.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-24.ux.1.1 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 48.288.7-24.vd.1.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-24.vf.1.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-24.xx.1.5 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 48.288.7-24.xz.1.5 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.7-24.yf.1.9 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 48.288.7-24.yh.1.7 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 48.288.8-48.dg.1.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.dg.1.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.dg.2.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.dg.2.3 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.dh.1.6 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.dh.1.14 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.dh.2.13 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.dh.2.14 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.di.1.7 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.di.1.15 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.di.2.11 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.di.2.12 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.dj.1.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.dj.1.9 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.dj.2.1 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-48.dj.2.2 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gk.1.5 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gk.1.7 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gk.2.6 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gk.2.8 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gl.1.11 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gl.1.12 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gl.2.3 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gl.2.4 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gm.1.11 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gm.1.12 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gm.2.3 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gm.2.4 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gn.1.6 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gn.1.8 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gn.2.5 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.8-24.gn.2.6 | $48$ | $2$ | $2$ | $8$ | $1$ | $2^{2}$ |
| 48.288.9-48.cn.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.cn.1.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 48.288.9-48.cp.1.5 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{5}$ |
| 48.288.9-48.cp.1.13 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{5}$ |
| 48.288.9-48.gw.1.19 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.gw.1.27 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.gx.1.10 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.gx.1.14 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.jy.1.21 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.jy.1.29 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.jz.1.11 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.jz.1.15 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.lt.1.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.lt.1.9 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 48.288.9-48.lv.1.1 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{5}$ |
| 48.288.9-48.lv.1.9 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{5}$ |
| 144.432.16-72.fb.1.12 | $144$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 240.288.7-120.dfo.1.6 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dfq.1.6 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dfw.1.6 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dfy.1.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dmm.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dmo.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dmu.1.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.7-120.dmw.1.13 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.288.8-240.ee.1.3 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ee.1.35 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ee.2.9 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ee.2.11 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ef.1.11 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ef.1.27 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ef.2.25 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.ef.2.26 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.eg.1.13 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.eg.1.29 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.eg.2.21 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.eg.2.22 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.eh.1.5 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.eh.1.37 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.eh.2.5 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-240.eh.2.6 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qo.1.2 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qo.1.10 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qo.2.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qo.2.8 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qp.1.23 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qp.1.24 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qp.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qp.2.8 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qq.1.23 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qq.1.24 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qq.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qq.2.8 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qr.1.4 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qr.1.12 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qr.2.2 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.8-120.qr.2.6 | $240$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 240.288.9-240.ic.1.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ic.1.21 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.id.1.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.id.1.41 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ji.1.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ji.1.43 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.jj.1.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.jj.1.22 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mo.1.13 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mo.1.45 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mp.1.7 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.mp.1.23 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ok.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ok.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ol.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.ol.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |