Invariants
Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
Index: | $96$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{8}\cdot4^{4}\cdot16^{4}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
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: | 16M1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.1.1898 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&30\\28&7\end{bmatrix}$, $\begin{bmatrix}29&2\\32&9\end{bmatrix}$, $\begin{bmatrix}43&8\\12&37\end{bmatrix}$, $\begin{bmatrix}45&29\\28&21\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 48.192.1-48.dh.2.1, 48.192.1-48.dh.2.2, 48.192.1-48.dh.2.3, 48.192.1-48.dh.2.4, 48.192.1-48.dh.2.5, 48.192.1-48.dh.2.6, 48.192.1-48.dh.2.7, 48.192.1-48.dh.2.8, 96.192.1-48.dh.2.1, 96.192.1-48.dh.2.2, 96.192.1-48.dh.2.3, 96.192.1-48.dh.2.4, 240.192.1-48.dh.2.1, 240.192.1-48.dh.2.2, 240.192.1-48.dh.2.3, 240.192.1-48.dh.2.4, 240.192.1-48.dh.2.5, 240.192.1-48.dh.2.6, 240.192.1-48.dh.2.7, 240.192.1-48.dh.2.8 |
Cyclic 48-isogeny field degree: | $8$ |
Cyclic 48-torsion field degree: | $64$ |
Full 48-torsion field degree: | $12288$ |
Jacobian
Conductor: | $2^{5}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} + 3 z^{2} - 2 w^{2} $ |
$=$ | $6 x^{2} - y^{2} + 4 w^{2}$ |
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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3}\cdot\frac{(81z^{8}-216z^{6}w^{2}+180z^{4}w^{4}-48z^{2}w^{6}+w^{8})^{3}}{w^{16}z^{2}(3z^{2}-4w^{2})(3z^{2}-2w^{2})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
16.48.1.u.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.0.bo.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0.ba.1 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0.bk.1 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0.bo.1 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.1.x.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.48.1.bm.2 | $48$ | $2$ | $2$ | $1$ | $0$ | 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 |
---|---|---|---|---|---|---|
48.288.17.bby.1 | $48$ | $3$ | $3$ | $17$ | $3$ | $1^{8}\cdot2^{4}$ |
48.384.17.xy.1 | $48$ | $4$ | $4$ | $17$ | $2$ | $1^{8}\cdot2^{4}$ |
96.192.9.gz.1 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
96.192.9.hh.1 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
96.192.9.kb.2 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
96.192.9.kj.1 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |