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: | $1$ | ||||||
| $\Q$-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: | 48.192.3.2668 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&6\\24&25\end{bmatrix}$, $\begin{bmatrix}5&16\\40&21\end{bmatrix}$, $\begin{bmatrix}17&28\\32&21\end{bmatrix}$, $\begin{bmatrix}27&16\\8&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.3.bn.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: | $2^{18}\cdot3^{2}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 64.2.b.a, 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ y t + z w + w t $ |
| $=$ | $x u + y z + y t - w t$ | |
| $=$ | $3 x t - w u$ | |
| $=$ | $3 x^{2} - y^{2} - w^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 24 x^{8} - 72 x^{6} y^{2} - 4 x^{6} z^{2} + 54 x^{4} y^{4} + 24 x^{4} y^{2} z^{2} - 27 x^{2} y^{4} z^{2} + 3 y^{4} z^{4} $ |
Geometric Weierstrass model Geometric Weierstrass model
| $ w^{2} $ | $=$ | $ 18 x^{2} y z - 12 y z^{3} $ |
| $0$ | $=$ | $3 x^{2} - y^{2} - z^{2}$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^2\,\frac{64512yw^{11}-127488yw^{9}u^{2}+58560yw^{7}u^{4}+32672yw^{5}u^{6}-21292yw^{3}u^{8}+30930ywu^{10}+828144zt^{11}+482112zt^{9}u^{2}-323676zt^{7}u^{4}-168156zt^{5}u^{6}-70929zt^{3}u^{8}+18432ztu^{10}-65536w^{12}+164352w^{10}u^{2}-115968w^{8}u^{4}+10528w^{6}u^{6}-10096w^{4}u^{8}+7150w^{2}u^{10}+1574640t^{12}-800928t^{10}u^{2}-564732t^{8}u^{4}-150660t^{6}u^{6}-32337t^{4}u^{8}+18747t^{2}u^{10}-1024u^{12}}{u^{4}(64yw^{7}-96yw^{5}u^{2}-4yw^{3}u^{4}-6ywu^{6}-324zt^{7}-108zt^{5}u^{2}+9zt^{3}u^{4}-32w^{6}u^{2}+16w^{4}u^{4}+10w^{2}u^{6}-324t^{8}+108t^{6}u^{2}+45t^{4}u^{4}-9t^{2}u^{6})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.3.bn.2 :
| $\displaystyle X$ | $=$ | $\displaystyle t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ 24X^{8}-72X^{6}Y^{2}+54X^{4}Y^{4}-4X^{6}Z^{2}+24X^{4}Y^{2}Z^{2}-27X^{2}Y^{4}Z^{2}+3Y^{4}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.2-16.d.2.9 | $16$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.96.0-24.ba.2.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-24.ba.2.8 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.1-48.a.1.6 | $48$ | $2$ | $2$ | $1$ | $1$ | $2$ |
| 48.96.1-48.a.1.16 | $48$ | $2$ | $2$ | $1$ | $1$ | $2$ |
| 48.96.2-16.d.2.8 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
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.z.1.4 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 48.384.5-48.bc.1.4 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 48.384.5-48.bw.2.4 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 48.384.5-48.bx.1.4 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 48.576.19-48.ja.2.11 | $48$ | $3$ | $3$ | $19$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 48.768.21-48.fc.2.4 | $48$ | $4$ | $4$ | $21$ | $3$ | $1^{8}\cdot2^{3}\cdot4$ |
| 240.384.5-240.ig.2.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ih.2.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.is.2.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.it.2.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |