Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $1152$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $8^{4}\cdot16^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16D5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.5.265 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}11&38\\20&31\end{bmatrix}$, $\begin{bmatrix}17&4\\16&33\end{bmatrix}$, $\begin{bmatrix}17&22\\20&9\end{bmatrix}$, $\begin{bmatrix}19&8\\8&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.5.bq.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $16$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{33}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 32.2.a.a, 1152.2.a.m, 1152.2.a.r, 1152.2.d.c |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y w + z t $ |
| $=$ | $6 x^{2} - z t$ | |
| $=$ | $8 y^{2} + 2 z^{2} - w^{2} - 2 t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} y^{2} + x^{4} z^{2} - 36 y^{4} z^{2} - 8 y^{2} z^{4} $ |
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 |
|---|
| $(0:1/2:0:0:1)$, $(0:-1/2:0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{64z^{12}+384z^{10}t^{2}+3264z^{8}t^{4}+5888z^{6}t^{6}+40128z^{4}t^{8}-73344z^{2}t^{10}+63w^{12}+576w^{8}t^{4}-2304w^{6}t^{6}+2304w^{4}t^{8}+18432w^{2}t^{10}+64t^{12}}{t^{4}(16z^{8}+32z^{6}t^{2}-48z^{4}t^{4}+128z^{2}t^{6}-w^{8}+4w^{6}t^{2}-4w^{4}t^{4}-32w^{2}t^{6})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.96.5.bq.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}Y^{2}+X^{4}Z^{2}-36Y^{4}Z^{2}-8Y^{2}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.i.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 48.96.1-8.i.2.4 | $48$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 48.96.3-48.e.2.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3-48.e.2.21 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3-48.f.2.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 48.96.3-48.f.2.21 | $48$ | $2$ | $2$ | $3$ | $1$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.384.9-48.da.1.12 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.dk.2.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 48.384.9-48.dr.1.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.dz.2.2 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 48.384.9-48.fk.2.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.fs.1.5 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 48.384.9-48.ga.3.4 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 48.384.9-48.go.1.14 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 48.576.21-48.ly.2.11 | $48$ | $3$ | $3$ | $21$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 48.768.25-48.gh.1.9 | $48$ | $4$ | $4$ | $25$ | $4$ | $1^{10}\cdot2^{3}\cdot4$ |
| 240.384.9-240.vh.1.15 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.vs.2.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.xs.1.15 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.yc.2.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bdl.2.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bdv.1.13 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bfw.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bgl.1.25 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |