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: | 16G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.1.925 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&30\\16&13\end{bmatrix}$, $\begin{bmatrix}13&17\\4&33\end{bmatrix}$, $\begin{bmatrix}15&41\\20&47\end{bmatrix}$, $\begin{bmatrix}41&9\\0&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.bx.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: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 99x - 378 $ |
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 |
|---|
| $(0:1:0)$, $(-6:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3}\cdot\frac{2088x^{2}y^{14}-3544571826x^{2}y^{12}z^{2}-6932242427688x^{2}y^{10}z^{4}+53011742336765199x^{2}y^{8}z^{6}+3003539166242014032x^{2}y^{6}z^{8}-276107538873630874545369x^{2}y^{4}z^{10}-318193573218156341529786828x^{2}y^{2}z^{12}-100516541203691949978889287705x^{2}z^{14}-1460844xy^{14}z-21002799096xy^{12}z^{3}+5451525458721xy^{10}z^{5}+772062233031073746xy^{8}z^{7}-567153163135642687704xy^{6}z^{9}-4110933851229296916667368xy^{4}z^{11}-3993284423003216365919256777xy^{2}z^{13}-1154460758489646977780788899690xz^{15}-y^{16}+347510736y^{14}z^{2}-841640347332y^{12}z^{4}+2668133519184168y^{10}z^{6}+4395738487801324788y^{8}z^{8}-16441666323193076500512y^{6}z^{10}-31815568820996942695406586y^{4}z^{12}-18253375230460820114149604352y^{2}z^{14}-3308169067604971667727148577241z^{16}}{y^{2}(x^{2}y^{12}+482652x^{2}y^{10}z^{2}+10805706018x^{2}y^{8}z^{4}+52918845228576x^{2}y^{6}z^{6}+82536342042418875x^{2}y^{4}z^{8}+38894063026481852508x^{2}y^{2}z^{10}+387420489x^{2}z^{12}+144xy^{12}z+17256078xy^{10}z^{3}+228725426700xy^{8}z^{5}+834645274821882xy^{6}z^{7}+1079026475131915020xy^{4}z^{9}+446709257654181508377xy^{2}z^{11}-2324522934xz^{13}+10224y^{12}z^{2}+456711696y^{10}z^{4}+3350448126639y^{8}z^{6}+7394410228074696y^{6}z^{8}+5727239278090175052y^{4}z^{10}+1280069276973550322256y^{2}z^{12}-24407490807z^{14})}$ |
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.e.2.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bz.2.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-16.e.2.12 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-24.bz.2.5 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.1-48.b.1.11 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-48.b.1.12 | $48$ | $2$ | $2$ | $1$ | $0$ | 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.r.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.y.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.bg.1.9 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ca.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.dm.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.dx.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.eb.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.eo.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.jh.1.5 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bfq.2.6 | $48$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 240.192.1-240.om.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ou.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ps.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.qa.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.tk.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ts.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.uq.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.uy.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.480.17-240.fb.2.4 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |