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: | 48.96.1.974 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}11&23\\44&27\end{bmatrix}$, $\begin{bmatrix}25&4\\36&47\end{bmatrix}$, $\begin{bmatrix}43&34\\12&13\end{bmatrix}$, $\begin{bmatrix}47&27\\24&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.cb.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 |
|---|
| $(6:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3}\cdot\frac{2232x^{2}y^{14}-69801223986x^{2}y^{12}z^{2}+8630998350150888x^{2}y^{10}z^{4}-141956909380425624561x^{2}y^{8}z^{6}+717926339135135488089168x^{2}y^{6}z^{8}-1475046256649715216484595289x^{2}y^{4}z^{10}+1303320875901571860504901151148x^{2}y^{2}z^{12}-411715752770322226985025083340825x^{2}z^{14}-1771884xy^{14}z+4867802805816xy^{12}z^{3}-266672882927098719xy^{10}z^{5}+2969872480609148627694xy^{8}z^{7}-11869594400810698466733144xy^{6}z^{9}+20790582672642410716129086408xy^{4}z^{11}-16356492996621153306933527814537xy^{2}z^{13}+4728671266773594021761143967723370xz^{15}-y^{16}+580010544y^{14}z^{2}-231707043048132y^{12}z^{4}+6293018435817973752y^{10}z^{6}-43829965684618689177612y^{8}z^{8}+118016006426462224514796192y^{6}z^{10}-141641803611311180361836796666y^{4}z^{12}+74765824943967299766167023760352y^{2}z^{14}-13550260500909963959106243235607001z^{16}}{y^{2}(x^{2}y^{12}-108x^{2}y^{10}z^{2}-10206x^{2}y^{8}z^{4}+314928x^{2}y^{6}z^{6}+90876411x^{2}y^{4}z^{8}+401769396x^{2}y^{2}z^{10}+387420489x^{2}z^{12}+1134xy^{10}z^{3}-218700xy^{8}z^{5}+6259194xy^{6}z^{7}+580333572xy^{4}z^{9}+2453663097xy^{2}z^{11}+2324522934xz^{13}-144y^{12}z^{2}+27216y^{10}z^{4}-1161297y^{8}z^{6}+35429400y^{6}z^{8}-5299529652y^{4}z^{10}-24794911296y^{2}z^{12}-24407490807z^{14})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.0-8.bb.2.8 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-8.bb.2.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.f.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.f.1.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.1-48.b.1.11 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-48.b.1.17 | $48$ | $2$ | $2$ | $1$ | $0$ | 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.i.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ba.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.bn.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.by.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.dm.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ea.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ee.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.eo.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.jt.1.17 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bfu.2.3 | $48$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 240.192.1-240.oq.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.oy.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.pw.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.qe.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.to.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.tw.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.uu.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.vc.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.480.17-240.fn.1.2 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |