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: | 48.96.1.1222 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&5\\36&7\end{bmatrix}$, $\begin{bmatrix}11&12\\44&5\end{bmatrix}$, $\begin{bmatrix}19&26\\8&47\end{bmatrix}$, $\begin{bmatrix}39&31\\4&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.bo.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 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2\cdot3}\cdot\frac{4176x^{2}y^{14}+56713149216x^{2}y^{12}z^{2}-887327030744064x^{2}y^{10}z^{4}-54284024152847563776x^{2}y^{8}z^{6}+24604992849854578950144x^{2}y^{6}z^{8}+18094983667622272994205302784x^{2}y^{4}z^{10}-166825072115400751987968876478464x^{2}y^{2}z^{12}+421596930836809960564255254978232320x^{2}z^{14}+5843376xy^{14}z-672089571072xy^{12}z^{3}-1395590517432576xy^{10}z^{5}+1581183453247639031808xy^{8}z^{7}+9292237424814369795342336xy^{6}z^{9}-538828321748326405461425258496xy^{4}z^{11}+4187262207135020604110150594199552xy^{2}z^{13}-9684318754352320554987748010250731520xz^{15}+y^{16}+2780085888y^{14}z^{2}+53864982229248y^{12}z^{4}+1366084361822294016y^{10}z^{6}-18004944846034226331648y^{8}z^{8}-538760522078390730768777216y^{6}z^{10}+8340260473011422545944664080384y^{4}z^{12}-38280102371311369824029071066005504y^{2}z^{14}+55501867011727212343338600744464941056z^{16}}{y^{2}(x^{2}y^{12}-3861216x^{2}y^{10}z^{2}+691565185152x^{2}y^{8}z^{4}-27094448757030912x^{2}y^{6}z^{6}+338068857005747712000x^{2}y^{4}z^{8}-1274480657251757342982144x^{2}y^{2}z^{10}+101559956668416x^{2}z^{12}-288xy^{12}z+276097248xy^{10}z^{3}-29276854617600xy^{8}z^{5}+854676761417607168xy^{6}z^{7}-8839384884280647843840xy^{4}z^{9}+29275537909624439332995072xy^{2}z^{11}+1218719480020992xz^{13}+40896y^{12}z^{2}-14614774272y^{10}z^{4}+857714720419584y^{8}z^{6}-15143752147096977408y^{6}z^{8}+93835088332229428051968y^{4}z^{10}-167781240271477187838738432y^{2}z^{12}-25593109080440832z^{14})}$ |
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.2.9 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.by.2.16 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-16.e.2.9 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-24.by.2.11 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.1-48.a.1.11 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.48.1-48.a.1.16 | $48$ | $2$ | $2$ | $1$ | $1$ | 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.m.1.5 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.u.1.4 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.bf.1.1 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.bv.2.3 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.dq.1.10 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.dt.1.7 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.ef.1.5 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.192.1-48.ek.1.2 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.288.9-48.iq.1.3 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bfh.2.7 | $48$ | $4$ | $4$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 240.192.1-240.nv.1.11 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.od.1.12 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.pb.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.pj.1.11 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.st.1.11 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.tb.1.13 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.tz.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.uh.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.480.17-240.ek.2.26 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |