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: | 48.192.3.2662 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&36\\40&5\end{bmatrix}$, $\begin{bmatrix}11&20\\16&5\end{bmatrix}$, $\begin{bmatrix}17&2\\24&41\end{bmatrix}$, $\begin{bmatrix}39&16\\8&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.3.bq.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 $ |
| $=$ | $3 x t + w u$ | |
| $=$ | $ - x u + 2 y z + 2 y t + 2 w t$ | |
| $=$ | $3 x z + y u - w u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 12 x^{8} - 36 x^{6} y^{2} + 4 x^{6} z^{2} + 27 x^{4} y^{4} - 24 x^{4} y^{2} z^{2} + 27 x^{2} y^{4} z^{2} + 6 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 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^2\,\frac{2064384yw^{11}+2039808yw^{9}u^{2}+468480yw^{7}u^{4}-130688yw^{5}u^{6}-42584yw^{3}u^{8}-30930ywu^{10}-26500608zt^{11}+7713792zt^{9}u^{2}+2589408zt^{7}u^{4}-672624zt^{5}u^{6}+141858zt^{3}u^{8}+18432ztu^{10}+2097152w^{12}+2629632w^{10}u^{2}+927744w^{8}u^{4}+42112w^{6}u^{6}+20192w^{4}u^{8}+7150w^{2}u^{10}-50388480t^{12}-12814848t^{10}u^{2}+4517856t^{8}u^{4}-602640t^{6}u^{6}+64674t^{4}u^{8}+18747t^{2}u^{10}+512u^{12}}{u^{4}(512yw^{7}+384yw^{5}u^{2}-8yw^{3}u^{4}+6ywu^{6}+2592zt^{7}-432zt^{5}u^{2}-18zt^{3}u^{4}-128w^{6}u^{2}-32w^{4}u^{4}+10w^{2}u^{6}+2592t^{8}+432t^{6}u^{2}-90t^{4}u^{4}-9t^{2}u^{6})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.3.bq.2 :
| $\displaystyle X$ | $=$ | $\displaystyle t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}u$ |
Equation of the image curve:
| $0$ | $=$ | $ 12X^{8}-36X^{6}Y^{2}+27X^{4}Y^{4}+4X^{6}Z^{2}-24X^{4}Y^{2}Z^{2}+27X^{2}Y^{4}Z^{2}+6Y^{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.1.10 | $16$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.96.0-24.bc.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-24.bc.1.8 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.1-48.a.1.6 | $48$ | $2$ | $2$ | $1$ | $1$ | $2$ |
| 48.96.1-48.a.1.20 | $48$ | $2$ | $2$ | $1$ | $1$ | $2$ |
| 48.96.2-16.d.1.5 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
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.bh.1.2 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 48.384.5-48.bk.1.4 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 48.384.5-48.by.1.4 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 48.384.5-48.bz.1.2 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 48.576.19-48.jd.1.10 | $48$ | $3$ | $3$ | $19$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 48.768.21-48.ff.1.4 | $48$ | $4$ | $4$ | $21$ | $3$ | $1^{8}\cdot2^{3}\cdot4$ |
| 240.384.5-240.iq.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ir.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.jc.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.jd.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |