Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $576$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $4^{8}\cdot16^{4}$ | Cusp orbits | $2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16J3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.3.1252 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}3&28\\40&37\end{bmatrix}$, $\begin{bmatrix}17&24\\16&13\end{bmatrix}$, $\begin{bmatrix}17&34\\24&41\end{bmatrix}$, $\begin{bmatrix}37&16\\32&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.3.ce.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{18}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 64.2.a.a, 576.2.d.a |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ - x u + z t $ |
| $=$ | $ - x u - y u + z w$ | |
| $=$ | $x w - x t - y t$ | |
| $=$ | $2 x y - 2 y^{2} + w t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} y^{4} + 8 x^{4} y^{2} z^{2} + 8 x^{4} z^{4} + 9 x^{2} y^{4} z^{2} + 24 x^{2} y^{2} z^{4} + \cdots + 9 y^{4} z^{4} $ |
Geometric Weierstrass model Geometric Weierstrass model
| $ w^{2} $ | $=$ | $ 6 x^{2} y z + 4 y z^{3} $ |
| $0$ | $=$ | $3 x^{2} + y^{2} + z^{2}$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2\,\frac{786240y^{2}z^{10}+1572480y^{2}z^{8}u^{2}+187920y^{2}z^{6}u^{4}-38340y^{2}z^{2}u^{8}-115560y^{2}u^{10}+65536z^{12}+196608z^{10}u^{2}+81912z^{8}u^{4}-32768z^{6}u^{6}+24216z^{4}u^{8}-15660z^{2}u^{10}+5971968wt^{11}+995328wt^{9}u^{2}+1741824wt^{7}u^{4}+156384wt^{3}u^{8}-26064wtu^{10}-5971968t^{12}-2488320t^{10}u^{2}-1990656t^{8}u^{4}-463104t^{6}u^{6}-156384t^{4}u^{8}+1413t^{2}u^{10}+1024u^{12}}{u^{4}(24y^{2}z^{6}-30y^{2}z^{2}u^{4}+48y^{2}u^{6}+z^{4}u^{4}-2z^{2}u^{6}-5184wt^{7}+180wt^{3}u^{4}-30wtu^{6}+5184t^{8}+1296t^{6}u^{2}-180t^{4}u^{4}-21t^{2}u^{6})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.3.ce.2 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{3}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}u$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}Y^{4}+8X^{4}Y^{2}Z^{2}+9X^{2}Y^{4}Z^{2}+8X^{4}Z^{4}+24X^{2}Y^{2}Z^{4}+9Y^{4}Z^{4}+12X^{2}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.1-16.a.2.4 | $16$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 24.96.0-24.bc.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-24.bc.1.7 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.1-16.a.2.10 | $48$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 48.96.2-48.d.1.6 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.96.2-48.d.1.19 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.384.5-48.bg.1.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 48.384.5-48.bi.1.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 48.384.5-48.ed.1.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 48.384.5-48.ee.1.2 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 48.576.19-48.ma.1.9 | $48$ | $3$ | $3$ | $19$ | $1$ | $1^{8}\cdot2^{2}\cdot4$ |
| 48.768.21-48.jh.2.6 | $48$ | $4$ | $4$ | $21$ | $1$ | $1^{8}\cdot2^{3}\cdot4$ |
| 96.384.9-96.q.3.9 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.q.4.9 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.r.3.3 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.r.4.5 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.bp.3.9 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.bp.4.9 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.bq.3.3 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.bq.4.5 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.5-240.sp.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.sq.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.wx.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.wy.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |