Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $2304$ | ||
| Index: | $96$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $8^{4}\cdot16^{4}$ | Cusp orbits | $4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $3$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16B5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.5.498 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}9&20\\38&31\end{bmatrix}$, $\begin{bmatrix}21&25\\20&3\end{bmatrix}$, $\begin{bmatrix}25&14\\6&47\end{bmatrix}$, $\begin{bmatrix}25&45\\44&31\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 48-isogeny field degree: | $32$ |
| Cyclic 48-torsion field degree: | $512$ |
| Full 48-torsion field degree: | $12288$ |
Jacobian
| Conductor: | $2^{38}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 256.2.a.b, 256.2.a.e, 576.2.a.c, 2304.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 3 x y + 2 z^{2} $ |
| $=$ | $3 y^{2} - 3 w^{2} - t^{2}$ | |
| $=$ | $8 x^{2} - y^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{8} + 2 x^{7} z - 2 x^{6} z^{2} + 20 x^{5} z^{3} - 36 x^{4} y^{4} + 40 x^{4} z^{4} - 144 x^{3} y^{4} z + \cdots + 16 z^{8} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x-\frac{1}{6}t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}w-\frac{1}{12}t$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 48.48.3.eb.2 :
| $\displaystyle X$ | $=$ | $\displaystyle 2z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -2w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w+t$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{4}+Y^{4}-Y^{3}Z-3Y^{2}Z^{2}-4YZ^{3}-2Z^{4} $ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.3.cb.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.48.1.lw.1 | $24$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 48.48.1.ib.1 | $48$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 48.48.1.ie.1 | $48$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 48.48.3.bn.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.48.3.cg.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.48.3.eb.2 | $48$ | $2$ | $2$ | $3$ | $3$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.192.9.yr.1 | $48$ | $2$ | $2$ | $9$ | $4$ | $1^{4}$ |
| 48.192.9.ys.1 | $48$ | $2$ | $2$ | $9$ | $6$ | $1^{4}$ |
| 48.192.9.yt.1 | $48$ | $2$ | $2$ | $9$ | $5$ | $1^{4}$ |
| 48.192.9.yu.1 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 48.288.21.cqy.1 | $48$ | $3$ | $3$ | $21$ | $6$ | $1^{12}\cdot4$ |
| 48.384.25.bak.1 | $48$ | $4$ | $4$ | $25$ | $11$ | $1^{14}\cdot2^{3}$ |
| 240.192.9.drp.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.192.9.drq.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.192.9.drr.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.192.9.drs.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |