Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.1.2114 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}19&6\\4&41\end{bmatrix}$, $\begin{bmatrix}25&29\\16&31\end{bmatrix}$, $\begin{bmatrix}35&28\\12&41\end{bmatrix}$, $\begin{bmatrix}45&29\\20&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.bi.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $12288$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 12 x y + 3 y^{2} - w^{2} $ |
| $=$ | $24 x^{2} - 6 x y - z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} - 6 x^{2} y^{2} - 9 x^{2} z^{2} + 9 z^{4} $ |
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 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2\,\frac{6048y^{2}z^{10}+432y^{2}z^{8}w^{2}-340848y^{2}z^{6}w^{4}-3013416y^{2}z^{4}w^{6}-4719654y^{2}z^{2}w^{8}-786429y^{2}w^{10}-2048z^{12}-6144z^{10}w^{2}+18960z^{8}w^{4}+23968z^{6}w^{6}-43392z^{4}w^{8}+524640z^{2}w^{10}+131071w^{12}}{w^{2}z^{2}(24y^{2}z^{6}+132y^{2}z^{4}w^{2}+42y^{2}z^{2}w^{4}+3y^{2}w^{6}-64z^{6}w^{2}-68z^{4}w^{4}-16z^{2}w^{6}-w^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.48.1.bi.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}-6X^{2}Y^{2}-9X^{2}Z^{2}+9Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.1-16.a.1.14 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.0-24.by.2.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.f.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.f.1.14 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-24.by.2.15 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.1-16.a.1.9 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.192.1-48.a.1.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.t.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.bl.2.9 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.br.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cf.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cm.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cy.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cz.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.eu.1.17 | $48$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bav.2.3 | $48$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 240.192.1-240.hr.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.hv.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ih.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.il.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.iz.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jh.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kf.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kn.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.480.17-240.cs.1.2 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |