Invariants
Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $576$ | ||
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 | $2^{4}\cdot4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-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: | 48.96.1.956 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&2\\4&27\end{bmatrix}$, $\begin{bmatrix}3&44\\8&39\end{bmatrix}$, $\begin{bmatrix}13&21\\4&37\end{bmatrix}$, $\begin{bmatrix}19&22\\40&35\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 48.192.1-48.ds.1.1, 48.192.1-48.ds.1.2, 48.192.1-48.ds.1.3, 48.192.1-48.ds.1.4, 48.192.1-48.ds.1.5, 48.192.1-48.ds.1.6, 48.192.1-48.ds.1.7, 48.192.1-48.ds.1.8, 96.192.1-48.ds.1.1, 96.192.1-48.ds.1.2, 96.192.1-48.ds.1.3, 96.192.1-48.ds.1.4, 240.192.1-48.ds.1.1, 240.192.1-48.ds.1.2, 240.192.1-48.ds.1.3, 240.192.1-48.ds.1.4, 240.192.1-48.ds.1.5, 240.192.1-48.ds.1.6, 240.192.1-48.ds.1.7, 240.192.1-48.ds.1.8 |
Cyclic 48-isogeny field degree: | $4$ |
Cyclic 48-torsion field degree: | $64$ |
Full 48-torsion field degree: | $12288$ |
Jacobian
Conductor: | $2^{6}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 4 x^{2} + 5 y^{2} - z^{2} - w^{2} $ |
$=$ | $8 x^{2} - 2 y^{2} + z^{2}$ |
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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2}{3}\cdot\frac{(81z^{8}+3240z^{6}w^{2}+4824z^{4}w^{4}+1440z^{2}w^{6}+16w^{8})^{3}}{w^{2}z^{2}(3z^{2}-2w^{2})^{8}(3z^{2}+2w^{2})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
16.48.0.x.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.48.0.bl.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0.x.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0.bu.1 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.1.bt.1 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.48.1.bu.2 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.48.1.cf.1 | $48$ | $2$ | $2$ | $1$ | $1$ | 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.ckh.2 | $48$ | $3$ | $3$ | $17$ | $2$ | $1^{8}\cdot2^{4}$ |
48.384.17.bld.2 | $48$ | $4$ | $4$ | $17$ | $3$ | $1^{8}\cdot2^{4}$ |
96.192.5.de.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5.dm.1 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5.fq.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.192.5.fy.1 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |