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: | 48.96.1.869 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&2\\36&7\end{bmatrix}$, $\begin{bmatrix}7&43\\12&31\end{bmatrix}$, $\begin{bmatrix}21&38\\32&1\end{bmatrix}$, $\begin{bmatrix}31&4\\12&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.br.1 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8\cdot3^8}\cdot\frac{144x^{2}y^{14}+27975456x^{2}y^{12}z^{2}+963434456064x^{2}y^{10}z^{4}+11684823005021184x^{2}y^{8}z^{6}+57467495853256605696x^{2}y^{6}z^{8}+108585934077968253517824x^{2}y^{4}z^{10}+86999036281299180133023744x^{2}y^{2}z^{12}+25129135301054609190719324160x^{2}z^{14}+11376xy^{14}z+1138966272xy^{12}z^{3}+29958838619904xy^{10}z^{5}+312646212022136832xy^{8}z^{7}+1408048013277364617216xy^{6}z^{9}+2566541326568911914663936xy^{4}z^{11}+2019588178660930928872783872xy^{2}z^{13}+577230379243244028538430423040xz^{15}+y^{16}+611712y^{14}z^{2}+31542628608y^{12}z^{4}+520362477932544y^{10}z^{6}+3641896362461478912y^{8}z^{8}+11667652474126803664896y^{6}z^{10}+17632588701196107190370304y^{4}z^{12}+12425776585066118837472067584y^{2}z^{14}+3308169067571803000335904014336z^{16}}{z^{5}y^{2}(11052x^{2}y^{6}z+875017728x^{2}y^{4}z^{3}+14544641470464x^{2}y^{2}z^{5}+63717843778928640x^{2}z^{7}+xy^{8}+616464xy^{6}z^{2}+29727710208xy^{4}z^{4}+387781034950656xy^{2}z^{6}+1463634728721580032xz^{8}+144y^{8}z+24012288y^{6}z^{3}+602744758272y^{4}z^{5}+4380986324680704y^{2}z^{7}+8388247240493236224z^{9})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.0-8.bb.1.6 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-8.bb.1.4 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.e.2.1 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.e.2.14 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.1-48.a.1.17 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.48.1-48.a.1.19 | $48$ | $2$ | $2$ | $1$ | $1$ | 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.f.2.5 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.v.2.3 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.bi.1.10 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.bu.1.3 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.dq.2.3 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.dw.2.2 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.ei.2.3 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.ek.1.5 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.288.9-48.it.1.19 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bfk.1.18 | $48$ | $4$ | $4$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 240.192.1-240.ny.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.og.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.pe.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.pm.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.sw.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.te.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.uc.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.uk.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.480.17-240.en.1.4 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |