Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $576$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.1.1093 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&35\\24&43\end{bmatrix}$, $\begin{bmatrix}7&25\\8&13\end{bmatrix}$, $\begin{bmatrix}9&22\\28&47\end{bmatrix}$, $\begin{bmatrix}19&34\\12&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.bp.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| 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
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 396x - 3024 $ |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2\cdot3^2}\cdot\frac{144x^{2}y^{14}-29877984x^{2}y^{12}z^{2}+3421586064384x^{2}y^{10}z^{4}-7019709937815960576x^{2}y^{8}z^{6}-57120375286048493666304x^{2}y^{6}z^{8}+236156864454986300239970304x^{2}y^{4}z^{10}-274890497894311762852615028736x^{2}y^{2}z^{12}+102928938192580688367952267837440x^{2}z^{14}-14544xy^{14}z+17991113472xy^{12}z^{3}-2571483498045696xy^{10}z^{5}-220105323076723642368xy^{8}z^{7}-1101433099070134426927104xy^{6}z^{9}+5188818435801971885029195776xy^{4}z^{11}-6227671554843046892456056455168xy^{2}z^{13}+2364335633386795431420220019834880xz^{15}+y^{16}-3120768y^{14}z^{2}+1123561767168y^{12}z^{4}-179518228013592576y^{10}z^{6}-3440830218852579360768y^{8}z^{8}+1955708659347210781065216y^{6}z^{10}+20327697899449010897212145664y^{4}z^{12}-32204524649248942861882748829696y^{2}z^{14}+13550260500909930790438851991044096z^{16}}{zy^{2}(12060x^{2}y^{10}z+1577968128x^{2}y^{8}z^{3}+56716488707328x^{2}y^{6}z^{5}+827509228120793088x^{2}y^{4}z^{7}+5268744303613522624512x^{2}y^{2}z^{9}+12176669570456349195632640x^{2}z^{11}+xy^{12}+765072xy^{10}z^{2}+64012591872xy^{8}z^{4}+1825817140076544xy^{6}z^{6}+22945477495143960576xy^{4}z^{8}+131284894794440127086592xy^{2}z^{10}+279704952435646619537375232xz^{12}+144y^{12}z+35943264y^{10}z^{3}+1797901277184y^{8}z^{5}+34388360257222656y^{6}z^{7}+298457973959800651776y^{4}z^{9}+1164917259352026520289280y^{2}z^{11}+1603019011082045150277402624z^{13})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.0-16.e.1.5 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bz.2.12 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-16.e.1.12 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-24.bz.2.5 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.1-48.a.1.3 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.48.1-48.a.1.29 | $48$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 48.192.1-48.o.1.7 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.u.1.3 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.bf.2.5 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.bw.2.2 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.dp.2.3 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.du.1.3 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.eg.2.2 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.ej.2.3 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.288.9-48.ir.1.5 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bfi.2.10 | $48$ | $4$ | $4$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 240.192.1-240.nw.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.oe.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.pc.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.pk.2.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.su.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.tc.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ua.2.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ui.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.480.17-240.el.1.4 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |