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: | 48.96.1.808 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}3&32\\8&27\end{bmatrix}$, $\begin{bmatrix}11&20\\12&29\end{bmatrix}$, $\begin{bmatrix}15&23\\44&47\end{bmatrix}$, $\begin{bmatrix}29&21\\36&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.bw.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $32$ |
| 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 |
|---|
| $(0:1:0)$, $(6:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^2}\cdot\frac{72x^{2}y^{14}+1867374x^{2}y^{12}z^{2}+26731141128x^{2}y^{10}z^{4}+6855185486148399x^{2}y^{8}z^{6}-6972702061285216512x^{2}y^{6}z^{8}-3603467780380039981689x^{2}y^{4}z^{10}-524312015331862950997572x^{2}y^{2}z^{12}-24540171192307636348712985x^{2}z^{14}+3636xy^{14}z+562222296xy^{12}z^{3}+10044857414241xy^{10}z^{5}-107473302283556466xy^{8}z^{7}+67226141300667384456xy^{6}z^{9}+39587542997756743507608xy^{4}z^{11}+5939170412867590801673943xy^{2}z^{13}+281850771115636280944373610xz^{15}-y^{16}-390096y^{14}z^{2}-17555652612y^{12}z^{4}-350621539089048y^{10}z^{6}+840046440149555508y^{8}z^{8}+59683491801367516512y^{6}z^{10}-77544013593479198063706y^{4}z^{12}-15356314015030356818143248y^{2}z^{14}-807658463770743059542110681z^{16}}{zy^{2}(3015x^{2}y^{10}z-49311504x^{2}y^{8}z^{3}+221548784013x^{2}y^{6}z^{5}-404057240293356x^{2}y^{4}z^{7}+321578631812348793x^{2}y^{2}z^{9}-92900616229677957120x^{2}z^{11}+xy^{12}-95634xy^{10}z^{2}+1000196748xy^{8}z^{4}-3566049101712xy^{6}z^{6}+5601923216587881xy^{4}z^{8}-4006497033521732394xy^{2}z^{10}+1066989717238031843328xz^{12}-72y^{12}z+2246454y^{10}z^{3}-14046103728y^{8}z^{5}+33582383063694y^{6}z^{7}-36432858149389728y^{4}z^{9}+17775226735718178105y^{2}z^{11}-3057516119159784603648z^{13})}$ |
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.1.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.by.2.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-16.e.1.3 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-24.by.2.15 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.1-48.b.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-48.b.1.11 | $48$ | $2$ | $2$ | $1$ | $0$ | 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.p.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.y.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.bg.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.bz.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.dl.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.dy.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ec.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.en.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.jg.2.13 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bfp.1.5 | $48$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 240.192.1-240.ol.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ot.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.pr.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.pz.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.tj.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.tr.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.up.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ux.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.480.17-240.fa.1.2 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |