Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $288$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{4}\cdot16^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.48.1.115 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&34\\40&45\end{bmatrix}$, $\begin{bmatrix}7&16\\10&9\end{bmatrix}$, $\begin{bmatrix}9&32\\46&15\end{bmatrix}$, $\begin{bmatrix}31&29\\0&29\end{bmatrix}$, $\begin{bmatrix}45&35\\46&43\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 48-isogeny field degree: | $16$ |
| Cyclic 48-torsion field degree: | $256$ |
| Full 48-torsion field degree: | $24576$ |
Jacobian
| Conductor: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x y + x w - 2 y z $ |
| $=$ | $3 x^{2} + 2 y^{2} + 4 y w - 6 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 4 x^{3} z + 3 x^{2} y^{2} + 4 x^{2} z^{2} - 6 x y^{2} z - 3 y^{2} z^{2} - z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{209018880xz^{11}-19191158280xz^{9}w^{2}+24365372832xz^{7}w^{4}-6101139456xz^{5}w^{6}+478158336xz^{3}w^{8}-11613312xzw^{10}-3791390733y^{2}z^{10}+25624409778y^{2}z^{8}w^{2}-18558928872y^{2}z^{6}w^{4}+3584136528y^{2}z^{4}w^{6}-245256912y^{2}z^{2}w^{8}+5474336y^{2}w^{10}+24894760482yz^{10}w-30256633404yz^{8}w^{3}+2215407888yz^{6}w^{5}+996423840yz^{4}w^{7}-115608288yz^{2}w^{9}+3207232yw^{11}+295612416z^{12}-16096497147z^{10}w^{2}+11905159518z^{8}w^{4}+1851705576z^{6}w^{6}-1015011216z^{4}w^{8}+89986896z^{2}w^{10}-2267168w^{12}}{6804xz^{9}w^{2}-145476xz^{7}w^{4}+312984xz^{5}w^{6}-105696xz^{3}w^{8}+6720xzw^{10}+243y^{2}z^{10}-29484y^{2}z^{8}w^{2}+234063y^{2}z^{6}w^{4}-265950y^{2}z^{4}w^{6}+61296y^{2}z^{2}w^{8}-3168y^{2}w^{10}-7290yz^{10}w+180468yz^{8}w^{3}-408726yz^{6}w^{5}+70020yz^{4}w^{7}+19680yz^{2}w^{9}-1856yw^{11}+6561z^{10}w^{2}-123768z^{8}w^{4}+189945z^{6}w^{6}+9486z^{4}w^{8}-18672z^{2}w^{10}+1312w^{12}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0.bi.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.96.3.bb.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.es.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.fv.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.gp.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.sg.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.sh.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.ta.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.tb.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.wn.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.wo.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.wv.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.ww.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.yt.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.yu.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.yx.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.yy.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.5.ld.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.le.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.96.5.lh.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.96.5.li.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.96.5.pr.1 | $48$ | $2$ | $2$ | $5$ | $4$ | $1^{2}\cdot2$ |
| 48.96.5.ps.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 48.96.5.pz.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.96.5.qa.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 48.144.7.qm.1 | $48$ | $3$ | $3$ | $7$ | $2$ | $1^{6}$ |
| 48.192.11.ku.1 | $48$ | $4$ | $4$ | $11$ | $2$ | $1^{10}$ |
| 240.96.3.ewy.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.exe.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.exs.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.exy.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fef.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fer.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fft.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fgf.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fnv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fnw.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fol.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fom.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fsn.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fso.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fsv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fsw.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.5.dhh.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dhi.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dhp.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dhq.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dkx.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dky.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dln.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dlo.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.240.17.zl.1 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 240.288.17.xcp.1 | $240$ | $6$ | $6$ | $17$ | $?$ | not computed |