Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $1152$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{2}\cdot16^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16N3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.3.2577 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&13\\4&3\end{bmatrix}$, $\begin{bmatrix}19&31\\40&29\end{bmatrix}$, $\begin{bmatrix}37&12\\36&31\end{bmatrix}$, $\begin{bmatrix}39&10\\4&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.3.ku.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{19}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 32.2.a.a, 1152.2.k.b |
Models
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ 18 x^{4} + 3 x^{2} y^{2} + 3 x^{2} z^{2} - y^{3} z - 2 y^{2} z^{2} + y z^{3} $ |
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.
| Canonical model |
|---|
| $(0:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{8640x^{2}y^{20}z^{2}+138240x^{2}y^{19}z^{3}+907200x^{2}y^{18}z^{4}+3041280x^{2}y^{17}z^{5}+208500480x^{2}y^{16}z^{6}+1629849600x^{2}y^{15}z^{7}+4268194560x^{2}y^{14}z^{8}+3251128320x^{2}y^{13}z^{9}+50307454080x^{2}y^{12}z^{10}+50307454080x^{2}y^{10}z^{12}-3251128320x^{2}y^{9}z^{13}+4268194560x^{2}y^{8}z^{14}-1629849600x^{2}y^{7}z^{15}+208500480x^{2}y^{6}z^{16}-3041280x^{2}y^{5}z^{17}+907200x^{2}y^{4}z^{18}-138240x^{2}y^{3}z^{19}+8640x^{2}y^{2}z^{20}-y^{24}-24y^{23}z+468y^{22}z^{2}+10024y^{21}z^{3}-108354y^{20}z^{4}-2615112y^{19}z^{5}-1206908y^{18}z^{6}+75836664y^{17}z^{7}-113541615y^{16}z^{8}-2783380592y^{15}z^{9}-3286821720y^{14}z^{10}-2861842416y^{13}z^{11}-15769425308y^{12}z^{12}+2861842416y^{11}z^{13}-3286821720y^{10}z^{14}+2783380592y^{9}z^{15}-113541615y^{8}z^{16}-75836664y^{7}z^{17}-1206908y^{6}z^{18}+2615112y^{5}z^{19}-108354y^{4}z^{20}-10024y^{3}z^{21}+468y^{2}z^{22}+24yz^{23}-z^{24}}{z^{2}y^{2}(y^{2}+2yz-z^{2})^{4}(12x^{2}y^{10}+96x^{2}y^{9}z+252x^{2}y^{8}z^{2}+192x^{2}y^{7}z^{3}+3000x^{2}y^{6}z^{4}+3000x^{2}y^{4}z^{6}-192x^{2}y^{3}z^{7}+252x^{2}y^{2}z^{8}-96x^{2}yz^{9}+12x^{2}z^{10}+y^{12}+8y^{11}z+46y^{10}z^{2}+216y^{9}z^{3}+751y^{8}z^{4}+208y^{7}z^{5}+1924y^{6}z^{6}-208y^{5}z^{7}+751y^{4}z^{8}-216y^{3}z^{9}+46y^{2}z^{10}-8yz^{11}+z^{12})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.1-16.v.2.1 | $16$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 48.96.1-16.v.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.384.5-48.fh.1.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.fx.1.7 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.gt.3.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.ii.1.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.iv.1.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.iy.1.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.je.2.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.jj.1.7 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.576.19-48.cnx.2.3 | $48$ | $3$ | $3$ | $19$ | $1$ | $1^{4}\cdot2^{2}\cdot8$ |
| 48.768.21-48.bdo.2.13 | $48$ | $4$ | $4$ | $21$ | $0$ | $1^{4}\cdot2^{3}\cdot8$ |
| 96.384.9-96.hl.1.9 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.hr.1.13 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.hv.1.1 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.hx.2.5 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.ib.1.13 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.ih.1.15 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.jb.1.9 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.jd.2.11 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.kp.2.6 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.kr.1.8 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.ll.1.2 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.lr.1.4 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.lv.2.12 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.lx.1.16 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.mb.1.4 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.mh.1.8 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.5-240.cdx.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cef.1.13 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cen.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cev.1.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cff.1.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cfv.1.15 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cgl.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.chb.1.13 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |