Invariants
| Level: | $48$ | $\SL_2$-level: | $8$ | Newform level: | $2304$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $8^{6}$ | Cusp orbits | $1^{2}\cdot4$ | ||
| Elliptic points: | $4$ 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: | 8H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.48.1.117 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}17&47\\24&19\end{bmatrix}$, $\begin{bmatrix}29&6\\32&17\end{bmatrix}$, $\begin{bmatrix}39&35\\16&1\end{bmatrix}$, $\begin{bmatrix}39&38\\2&25\end{bmatrix}$, $\begin{bmatrix}45&7\\38&43\end{bmatrix}$ |
| 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: | $2^{8}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 2304.2.a.h |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 30x + 56 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^4}\cdot\frac{96x^{2}y^{14}-2825928x^{2}y^{12}z^{2}+4011762816x^{2}y^{10}z^{4}+5432574134880x^{2}y^{8}z^{6}-418322601225216x^{2}y^{6}z^{8}-1367606740155588864x^{2}y^{4}z^{10}-125040613889530810368x^{2}y^{2}z^{12}+57706377064481401288704x^{2}z^{14}-4440xy^{14}z+46339776xy^{12}z^{3}-13870163952xy^{10}z^{5}-55851722453376xy^{8}z^{7}-9558645234978816xy^{6}z^{9}+8744717451061075968xy^{4}z^{11}+1444548215140906466304xy^{2}z^{13}-360240179884312886452224xz^{15}-y^{16}+131136y^{14}z^{2}-501191712y^{12}z^{4}-319590614016y^{10}z^{6}+266022891346176y^{8}z^{8}+169372777889710080y^{6}z^{10}-7218061173726520320y^{4}z^{12}-9411478406749928669184y^{2}z^{14}+517658685348717743763456z^{16}}{z^{2}(x^{2}y^{12}-77328x^{2}y^{10}z^{2}+259226568x^{2}y^{8}z^{4}-212034723840x^{2}y^{6}z^{6}+66252547178496x^{2}y^{4}z^{8}-8752963672866816x^{2}y^{2}z^{10}+412282644095434752x^{2}z^{12}-80xy^{12}z+1463940xy^{10}z^{3}-2980735200xy^{8}z^{5}+1887350906880xy^{6}z^{7}-502540677218304xy^{4}z^{9}+59385303921917952xy^{2}z^{11}-2573732421890998272xz^{13}+3076y^{12}z^{2}-20438784y^{10}z^{4}+23872860432y^{8}z^{6}-9828091625472y^{6}z^{8}+1759467040800768y^{4}z^{10}-137745393081384960y^{2}z^{12}+3698407382037037056z^{14})}$ |
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 |
| 48.24.0.l.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.24.1.g.2 | $48$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 48.96.3.ba.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.et.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.fu.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.gq.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.sf.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.sj.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.sw.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $1^{2}$ |
| 48.96.3.te.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.wk.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.wy.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.ys.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.yy.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.5.le.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.96.5.lg.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 48.96.5.pu.1 | $48$ | $2$ | $2$ | $5$ | $4$ | $1^{2}\cdot2$ |
| 48.96.5.pw.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 48.144.7.qh.1 | $48$ | $3$ | $3$ | $7$ | $2$ | $1^{2}\cdot2^{2}$ |
| 48.192.11.kr.1 | $48$ | $4$ | $4$ | $11$ | $7$ | $1^{8}\cdot2$ |
| 240.96.3.ewx.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.exd.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.exr.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.exx.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fec.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.feo.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.ffq.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fgc.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fno.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fok.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fsk.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fsu.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.5.dhg.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dhm.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dkw.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dlg.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.240.17.zj.1 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 240.288.17.xcm.1 | $240$ | $6$ | $6$ | $17$ | $?$ | not computed |