Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $288$ | ||
| 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: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.1.598 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}13&40\\24&25\end{bmatrix}$, $\begin{bmatrix}27&25\\4&19\end{bmatrix}$, $\begin{bmatrix}33&28\\8&21\end{bmatrix}$, $\begin{bmatrix}35&11\\32&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.ci.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $4$ |
| Cyclic 48-torsion field degree: | $32$ |
| Full 48-torsion field degree: | $12288$ |
Jacobian
| Conductor: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 36x $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3}\cdot\frac{2160x^{2}y^{14}+31748332320x^{2}y^{12}z^{2}+177742138337280x^{2}y^{10}z^{4}-21389675816365056x^{2}y^{8}z^{6}+1781413360312320000x^{2}y^{6}z^{8}-32164757659913748480x^{2}y^{4}z^{10}+224853744063873024000x^{2}y^{2}z^{12}-538990877234083921920x^{2}z^{14}+1613232xy^{14}z+1010496176640xy^{12}z^{3}+695941766348544xy^{10}z^{5}-97331165159915520xy^{8}z^{7}+2698938294890397696xy^{6}z^{9}-24953281353429811200xy^{4}z^{11}+74881781010929811456xy^{2}z^{13}+y^{16}+457228800y^{14}z^{2}+18179998048512y^{12}z^{4}+2716620727357440y^{10}z^{6}+3687495398572032y^{8}z^{8}+1515951619870556160y^{6}z^{10}+2887755807909740544y^{4}z^{12}+7897302230536028160y^{2}z^{14}+4738381338321616896z^{16}}{y^{2}(x^{2}y^{12}-21600x^{2}y^{10}z^{2}-44509824x^{2}y^{8}z^{4}-21888755712x^{2}y^{6}z^{6}+685686435840x^{2}y^{4}z^{8}+1438766052802560x^{2}y^{2}z^{10}+104098955585126400x^{2}z^{12}-72xy^{12}z-147744xy^{10}z^{3}-318007296xy^{8}z^{5}-337038465024xy^{6}z^{7}-120524084379648xy^{4}z^{9}-11552445071032320xy^{2}z^{11}+2016y^{12}z^{2}+5412096y^{10}z^{4}+5588082432y^{8}z^{6}+2179684712448y^{6}z^{8}+245436561948672y^{4}z^{10}-203119913336832y^{2}z^{12}+3656158440062976z^{14})}$ |
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.g.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bl.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-16.g.1.3 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-24.bl.1.3 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.1-48.b.1.11 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-48.b.1.26 | $48$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 48.192.1-48.eb.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.eb.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ec.1.6 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ec.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ed.1.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ed.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ee.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ee.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.lw.1.1 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{8}$ |
| 48.384.9-48.bgn.1.2 | $48$ | $4$ | $4$ | $9$ | $1$ | $1^{8}$ |
| 96.192.3-96.t.1.11 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.t.2.13 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.bp.1.5 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.bp.2.9 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.br.1.5 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.br.2.9 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.cl.1.5 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.cl.2.5 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.1-240.vv.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.vv.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.vw.1.13 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.vw.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.vx.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.vx.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.vy.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.vy.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.480.17-240.he.1.3 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |