Invariants
Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
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: | 48.96.1.1624 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&40\\32&29\end{bmatrix}$, $\begin{bmatrix}7&35\\0&41\end{bmatrix}$, $\begin{bmatrix}23&38\\24&11\end{bmatrix}$, $\begin{bmatrix}43&44\\0&35\end{bmatrix}$, $\begin{bmatrix}47&31\\16&13\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 48.192.1-48.bl.1.1, 48.192.1-48.bl.1.2, 48.192.1-48.bl.1.3, 48.192.1-48.bl.1.4, 48.192.1-48.bl.1.5, 48.192.1-48.bl.1.6, 48.192.1-48.bl.1.7, 48.192.1-48.bl.1.8, 48.192.1-48.bl.1.9, 48.192.1-48.bl.1.10, 48.192.1-48.bl.1.11, 48.192.1-48.bl.1.12, 240.192.1-48.bl.1.1, 240.192.1-48.bl.1.2, 240.192.1-48.bl.1.3, 240.192.1-48.bl.1.4, 240.192.1-48.bl.1.5, 240.192.1-48.bl.1.6, 240.192.1-48.bl.1.7, 240.192.1-48.bl.1.8, 240.192.1-48.bl.1.9, 240.192.1-48.bl.1.10, 240.192.1-48.bl.1.11, 240.192.1-48.bl.1.12 |
Cyclic 48-isogeny field degree: | $8$ |
Cyclic 48-torsion field degree: | $128$ |
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 $ | $=$ | $ 3 x^{2} - 3 z^{2} - w^{2} $ |
$=$ | $3 x^{2} - 8 y^{2} + 3 z^{2}$ |
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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{2^2}{3}\cdot\frac{(1296z^{8}+864z^{6}w^{2}-360z^{4}w^{4}-168z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{2}(3z^{2}+w^{2})(6z^{2}+w^{2})^{8}}$ |
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 |
---|---|---|---|---|---|---|
16.48.1.g.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.0.be.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0.k.1 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0.by.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.0.bz.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.48.1.bi.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.48.1.bj.1 | $48$ | $2$ | $2$ | $1$ | $0$ | 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 |
---|---|---|---|---|---|---|
48.192.5.hj.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.192.5.hk.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.192.5.hl.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.192.5.hm.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
48.288.17.ns.1 | $48$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
48.384.17.pk.1 | $48$ | $4$ | $4$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
240.192.5.bye.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.192.5.byf.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.192.5.byg.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.192.5.byh.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |