Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $96$ | ||
| 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 $4$ are rational) | Cusp widths | $1^{2}\cdot2^{3}\cdot3^{2}\cdot6^{3}\cdot16\cdot48$ | Cusp orbits | $1^{4}\cdot2^{4}$ | ||
| 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: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48L3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.3.584 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&13\\12&37\end{bmatrix}$, $\begin{bmatrix}17&8\\0&1\end{bmatrix}$, $\begin{bmatrix}17&8\\36&47\end{bmatrix}$, $\begin{bmatrix}19&44\\0&7\end{bmatrix}$, $\begin{bmatrix}41&19\\0&19\end{bmatrix}$, $\begin{bmatrix}47&42\\12&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.3.qd.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $16$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{13}\cdot3^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 24.2.a.a, 96.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ 2 x^{4} + x^{2} y^{2} + x^{2} z^{2} - y^{3} z + y z^{3} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:1)$, $(0:1:0)$, $(0:1:1)$, $(0:-1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{261415x^{2}y^{22}+2091320x^{2}y^{21}z+2875565x^{2}y^{20}z^{2}-7840400x^{2}y^{19}z^{3}-15942215x^{2}y^{18}z^{4}+5749080x^{2}y^{17}z^{5}-45792645x^{2}y^{16}z^{6}+39956160x^{2}y^{15}z^{7}+1180053990x^{2}y^{14}z^{8}+49102960x^{2}y^{13}z^{9}-6752746750x^{2}y^{12}z^{10}-323123680x^{2}y^{11}z^{11}+19194476530x^{2}y^{10}z^{12}-1249350800x^{2}y^{9}z^{13}-30576914730x^{2}y^{8}z^{14}+6168308160x^{2}y^{7}z^{15}+27458173635x^{2}y^{6}z^{16}-9206668200x^{2}y^{5}z^{17}-12784599295x^{2}y^{4}z^{18}+6099055600x^{2}y^{3}z^{19}+2101957765x^{2}y^{2}z^{20}-1528824520x^{2}yz^{21}+191102975x^{2}z^{22}+131072y^{24}+1311449y^{23}z+3150102y^{22}z^{2}-2882855y^{21}z^{3}-17066196y^{20}z^{4}-4465925y^{19}z^{5}+40368126y^{18}z^{6}+95404491y^{17}z^{7}-152428272y^{16}z^{8}-1003061302y^{15}z^{9}+1154466956y^{14}z^{10}+5151873738y^{13}z^{11}-5659385720y^{12}z^{12}-13718166522y^{11}z^{13}+15834861004y^{10}z^{14}+19769918118y^{9}z^{15}-25899307248y^{8}z^{16}-14287921299y^{7}z^{17}+24363252574y^{6}z^{18}+2852294125y^{5}z^{19}-12056081364y^{4}z^{20}+2101956335y^{3}z^{21}+2293234998y^{2}z^{22}-955514881yz^{23}+95551488z^{24}}{z^{3}y(y-z)^{2}(y+z)(729x^{2}y^{15}+2187x^{2}y^{14}z+2187x^{2}y^{13}z^{2}+729x^{2}y^{12}z^{3}+26973x^{2}y^{11}z^{4}+39799x^{2}y^{10}z^{5}-27817x^{2}y^{9}z^{6}-52867x^{2}y^{8}z^{7}+10667x^{2}y^{7}z^{8}+21249x^{2}y^{6}z^{9}-6079x^{2}y^{5}z^{10}-2709x^{2}y^{4}z^{11}+1631x^{2}y^{3}z^{12}-323x^{2}y^{2}z^{13}+29x^{2}yz^{14}-x^{2}z^{15}-729y^{16}z-729y^{15}z^{2}-2187y^{14}z^{3}+12393y^{13}z^{4}+39947y^{12}z^{5}+5843y^{11}z^{6}-57087y^{10}z^{7}-27267y^{9}z^{8}+26405y^{8}z^{9}+10837y^{7}z^{10}-7633y^{6}z^{11}-677y^{5}z^{12}+1129y^{4}z^{13}-271y^{3}z^{14}+27y^{2}z^{15}-yz^{16})}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.f.1.1 | $16$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.0-16.f.1.1 | $16$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.ir.1.8 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 48.96.1-24.ir.1.17 | $48$ | $2$ | $2$ | $1$ | $0$ | $2$ |
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.kd.3.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kd.4.9 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kh.1.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kh.2.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kl.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kl.4.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kp.1.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.kp.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.7-48.ck.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.cp.1.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.dc.1.33 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.de.1.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ed.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.eg.2.17 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.ek.2.25 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.em.1.17 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.fd.1.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fd.2.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fe.1.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fe.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fl.2.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fl.4.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fm.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fm.4.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fy.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fy.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gc.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gc.4.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gg.1.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gg.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gk.3.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gk.4.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gv.1.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.gw.1.7 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hd.1.13 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.he.2.9 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.ht.1.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hu.1.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ib.1.7 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ic.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bax.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bay.2.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbf.1.4 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbg.1.2 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfn.1.5 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfo.2.10 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfv.1.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfw.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bhn.3.5 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhn.4.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bho.2.7 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bho.4.13 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhv.1.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhv.2.5 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhw.1.4 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhw.2.7 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.576.13-48.ca.1.1 | $48$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{3}$ |
| 240.384.5-240.clr.3.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.clr.4.17 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.clv.3.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.clv.4.17 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.clz.2.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.clz.4.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cmd.1.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cmd.3.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.7-240.se.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.si.2.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.so.2.41 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ss.1.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ti.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.tm.2.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ts.2.41 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.tw.1.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vd.2.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vd.4.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ve.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ve.4.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vt.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vt.3.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vu.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vu.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ww.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ww.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xa.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xa.4.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xe.3.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xe.4.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xi.3.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xi.4.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.yn.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.yo.1.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zd.2.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ze.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.baj.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bak.1.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.baz.2.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bba.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.9-240.ftn.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fto.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fud.1.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fue.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fvj.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fvk.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fvz.1.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwa.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdr.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdr.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gds.1.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gds.3.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.geh.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.geh.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gei.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gei.4.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |