Invariants
Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $576$ | ||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (all of which are rational) | Cusp widths | $2^{2}\cdot4\cdot16$ | Cusp orbits | $1^{4}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 16A1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.48.1.252 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}11&29\\4&27\end{bmatrix}$, $\begin{bmatrix}17&39\\40&11\end{bmatrix}$, $\begin{bmatrix}19&47\\24&41\end{bmatrix}$, $\begin{bmatrix}35&42\\20&5\end{bmatrix}$, $\begin{bmatrix}37&14\\32&9\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 48.24.1.a.1 for the level structure with $-I$) |
Cyclic 48-isogeny field degree: | $8$ |
Cyclic 48-torsion field degree: | $64$ |
Full 48-torsion field degree: | $24576$ |
Jacobian
Conductor: | $2^{6}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 576.2.a.c |
Models
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} - 36x $ |
Rational points
This modular curve has infinitely many rational points, including 8 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^8}\cdot\frac{243x^{2}y^{4}z^{2}+36xy^{6}z+19683xy^{2}z^{5}+y^{8}+531441z^{8}}{z^{5}y^{2}x}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
16.24.0-8.n.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0-8.n.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
48.96.1-48.a.2.12 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.d.1.6 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.g.1.4 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.i.1.2 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bo.1.1 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bo.2.9 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bp.1.1 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bp.2.5 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bq.1.1 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bq.2.9 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.br.1.1 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.br.2.3 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bs.1.1 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bs.2.3 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bt.1.1 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bt.2.9 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bu.1.1 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bu.2.5 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bv.1.1 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.bv.2.9 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.cf.1.4 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.cg.1.6 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.cj.1.2 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.96.1-48.ck.1.2 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
48.144.5-48.e.1.49 | $48$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
48.192.5-48.op.1.2 | $48$ | $4$ | $4$ | $5$ | $2$ | $1^{4}$ |
240.96.1-240.cg.1.18 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.ch.1.18 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.ck.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.cl.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.cw.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.cw.2.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.cx.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.cx.2.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.cy.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.cy.2.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.cz.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.cz.2.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.da.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.da.2.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.db.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.db.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.dc.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.dc.2.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.dd.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.dd.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.ec.1.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.ed.1.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.eg.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.96.1-240.eh.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.240.9-240.a.1.4 | $240$ | $5$ | $5$ | $9$ | $?$ | not computed |
240.288.9-240.mi.1.2 | $240$ | $6$ | $6$ | $9$ | $?$ | not computed |
240.480.17-240.fq.1.36 | $240$ | $10$ | $10$ | $17$ | $?$ | not computed |