Invariants
Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $576$ | ||
Index: | $192$ | $\PSL_2$-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: | 48.192.1.311 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&2\\36&31\end{bmatrix}$, $\begin{bmatrix}13&16\\0&1\end{bmatrix}$, $\begin{bmatrix}25&30\\32&41\end{bmatrix}$, $\begin{bmatrix}29&24\\28&43\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 48.96.1.o.1 for the level structure with $-I$) |
Cyclic 48-isogeny field degree: | $8$ |
Cyclic 48-torsion field degree: | $32$ |
Full 48-torsion field degree: | $6144$ |
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 $ | $=$ | $ x z + 2 y^{2} + z^{2} $ |
$=$ | $6 x^{2} + 8 x z - 8 y^{2} + 8 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 6 y^{2} z^{2} + 4 z^{4} $ |
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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^4}\cdot\frac{(1296z^{8}+432z^{6}w^{2}+180z^{4}w^{4}+24z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{8}(6z^{2}+w^{2})^{4}(12z^{2}+w^{2})^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.1.o.1 :
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}w$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
$0$ | $=$ | $ X^{4}+6Y^{2}Z^{2}+4Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
16.96.0-16.d.2.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.96.0-24.bc.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.96.0-16.d.2.12 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.96.0-24.bc.1.5 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.96.0-48.bl.1.7 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.96.0-48.bl.1.10 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.96.0-48.br.2.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.96.0-48.br.2.15 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
48.96.1-48.a.2.9 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.a.2.14 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bp.2.2 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bp.2.15 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bv.1.7 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bv.1.10 | $48$ | $2$ | $2$ | $1$ | $1$ | 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 |
---|---|---|---|---|---|---|
48.384.5-48.bh.2.5 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
48.384.5-48.by.1.6 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
48.384.5-48.ee.1.4 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
48.384.5-48.ej.1.3 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
48.576.17-48.ej.1.1 | $48$ | $3$ | $3$ | $17$ | $2$ | $1^{8}\cdot2^{4}$ |
48.768.17-48.ik.2.11 | $48$ | $4$ | $4$ | $17$ | $3$ | $1^{8}\cdot2^{4}$ |
240.384.5-240.bfs.2.13 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.384.5-240.bfw.1.11 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.384.5-240.bgq.1.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.384.5-240.bgu.2.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |